Statistics for Psychology

Prepare a written response to the following assignments located in the text:

Ch. 2, Practice Problems: 11, 12, 13, 16, and 21

· Ch. 3, Practice Problems: 14, 15, 22, and 25

·
See instructions attached. Use the answer template provided in the attached to record and submit answers.

Please submit your answers to Chapter 2 Practice Problems 11, 12, 13, 16, & 21 in accordance with the attached instructions.

Please submit your answers to Chapter 3 Practice Problems 14, 15, 22, & 25 in accordance with the attached instructions.

Name:

Chapter 2 Instructions

Practice Problem 11, 12, 13, 16, & 21

Due Week 3 Day 6 (Sunday)

Follow the instructions below to submit your answers for Chapter 2 Practice Problem 11, 12, 13 16, & 21.

1. Save Chapter 2 Instructions to your computer.

2. Type your answers into the shaded boxes below. The boxes will expand as you type your answers.

3. Resave this form to your computer with your answers filled-in.

4. Attach the saved form to your reply when you turn-in your work in the Assignments section of the Classroom tab. Note: Each question in the assignments section will be listed separately; however, you only need to submit this form one time to turn-in your answers.

Below is an explanation of the symbols in Chapter 2.

M = Mean

Mdn. = Median

SS = Sum of Squared Deviations

SD2 = Variance

SD = Standard Deviation

Read each question in your text book and then type your answers for Chapter 2 Practice Problem 11, 12, 13, & 16 into the shaded boxes below. Please record only your answers. It is not necessary to show your work. Round your answers to 2 decimal places.

11. M =

Mdn. =

SS =

SD2 =

SD =

12. M =

Mdn. =
SS =
SD2 =
SD =

13. M =

Mdn. =
SS =
SD2 =
SD =

16a. Governors
–
M =
SD =

CEO’s
–
M =
SD =

16b. Explain your answer below:

16c. Explain below how the means and standard deviations differ:

21. Explain your answer below:

Central Tendency and Variability

Chapter Outline

✪ Central Tendency 3

✪ Variability 43

✪ Controversy: The Tyranny
of the Mean 5

✪ Central Tendency and Variability
in Research Articles 5

A
s we noted in Chapter 1, the purpose of descriptive statistics is to make a
group of scores understandable. We looked at some ways of getting that un-
derstanding through tables and graphs. In this chapter, we consider the main

statistical techniques for describing a group of scores with numbers. First, you can
describe a group of scores in terms of a representative (or typical) value, such as an
average. A representative value gives the central tendency of a group of scores. A
representative value is a simple way, with a single number, to describe a group of
scores (and there may be hundreds or even thousands of scores). The main represen-
tative value we consider is the mean. Next, we focus on ways of describing how
spread out the numbers are in a group of scores. In other words, we consider the
amount of variation, or variability, among the scores. The two measures of variabil-
ity you will learn about are called the variance and standard deviation

In this chapter, for the first time in this book, you will use statistical formulas.
Such formulas are not here to confuse you. Hopefully, you will come to see that they
actually simplify things and provide a very straightforward, concise way of describ-
ing statistical procedures. To help you grasp what such formulas mean in words,
whenever we present formulas in this book we always also give the “translation” in
ordinary English.

✪ Summary 5

✪ Key Terms 57

✪ Example Worked-Out Problems 57

✪ Practice Problems 5

✪ Using SPSS 62

✪ Chapter Notes 65

CHAPTER 2

T I P F O R S U C C E S S
Before beginning this chapter, you
should be sure you are comfort-
able with the key terms of variable,
score, and value that we consid-
ered in Chapter 1.

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34 Chapter 2

Central Tenden

The central tendency of a group of scores (a distribution) refers to the middle of the
group of scores. You will learn about three measures of central tendency: mean,
mode, and median. Each measure of central tendency uses its own method to come up
with a single number describing the middle of a group of scores. We start with the
mean, the most commonly used measure of central tendency. Understanding the mean
is also an important foundation for much of what you learn in later chapters.

The

Mean

Usually the best measure of central tendency is the ordinary average, the sum of all
the scores divided by the number of scores. In statistics, this is called the mean. The
average, or mean, of a group of scores is a representative value.

Suppose 10 students, as part of a research study, record the total number of
dreams they had during the last week. The numbers of dreams were as follows:

7, 8, 8, 7, 3, 1, 6, 9, 3,

The mean of these 10 scores is 6 (the sum of 60 dreams divided by 10 students).
That is, on the average, each student had 6 dreams in the past week. The information
for the 10 students is thus summarized by the single number 6.

You can think of the mean as a kind of balancing point for the distribution of
scores. Try it by visualizing a board balanced over a log, like a rudimentary teeter-
totter. Imagine piles of blocks set along the board according to their values, one f

each score in the distribution (like a histogram made of blocks). The mean is the
point on the board where the weight of the blocks on one side balances exactly with
the weight on the other side. Figure 2–1 shows this for the number of dreams for the
10 students.

Mathematically, you can think of the mean as the point at which the total distance
to all the scores above that point equals the total distance to all the scores below that
point. Let’s first figure the total distance from the mean to all the scores above the
mean for the dreams example shown in Figure 2–1. There are two scores of 7, each of
which is 1 unit above 6 (the mean). There are three scores of 8, each of which is 2
units above 6. And, there is one score of 9, which is 3 units above 6. This gives a total
distance of 11 units from the mean to all the scores above
the mean. Now, let’s look at the scores below the mean. There are two scores of 3,
each of which is 3 units below 6 (the mean). And there is one score of 1, which is 5
units below 6. This gives a total distance of 11 units from the mean to all
of the scores below the mean. Thus, you can see that the total distance from the mean
to the scores above the mean is the same as the total distance from the mean to the
scores below the mean. The scores above the mean balance out the scores below the
mean (and vice-versa).

(3 + 3 + 5)

(1 + 1 + 2 + 2 + 2 + 3)

mean arithmetic average of a group of
scores; sum of the scores divided by the
number of scores.

5 6 7 8 91 2 3 4

M = 6

Figure 2–1 Mean of the distribution of the number of dreams during a week for 10 students,
illustrated using blocks on a board balanced on a log.

central tendency typical or most
representative value of a group of scores.

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Central Tendency and Variability 35

Some other examples are shown in Figure 2–2. Notice that there doesn’t have to
be a block right at the balance point. That is, the mean doesn’t have to be a score ac-
tually in the distribution. The mean is the average of the scores, the balance point.
The mean can be a decimal number, even if all the scores in the distribution have to
be whole numbers (a mean of 2.30 children, for example). For each distribution in
Figure 2–2, the total distance from the mean to the scores above the mean is the same
as the total distance from the mean to the scores below the mean. (By the way, thi

analogy to blocks on a board, in reality, works out precisely only if the board has no
weight of its own.)

Formula for the Mean and Statistical Symbols
The rule for figuring the mean is to add up all the scores and divide by the number of
scores. Here is how this rule is written as a formula:

(2–1)

M is a symbol for the mean. An alternative symbol, (“X-bar”), is sometimes
used. However, M is almost always used in research articles in psychology, as rec-
ommended by the style guidelines of the American Psychological Association
(2001). You will see used mostly in advanced statistics books and in articles about
statistics. In fact, there is not a general agreement for many of the symbols used in
statistics. (In this book we generally use the symbols most widely found in psychol-
ogy research articles.)
S, the capital Greek letter sigma, is the symbol for “sum of.” It means “add up

all the numbers for whatever follows.” It is the most common special arithmetic
symbol used in statistics.

X stands for the scores in the distribution of the variable X. We could have
picked any letter. However, if there is only one variable, it is usually called X. In later
chapters we use formulas with more than one variable. In those formulas, we use a
second letter along with X (usually Y ) or subscripts (such as and ).

is “the sum of X.” This tells you to add up all the scores in the distribution
of the variable X. Suppose X is the number of dreams of our 10 students: X is

, which is 60.7 + 8 + 8 + 7 + 3 + 1 + 6 + 9 + 3 + 8
©

©X
X2X

X
X

M =
gX

M mean.

5 6 7 8 91 2 3 4
5 6 7 8 91 2 3 4
5 6 7 8 91 2 3 4
5 6 7 8 91 2 3 4
M = 6

M = 3.

M = 6
M = 6

Figure 2–2 Means of various distributions illustrated with blocks on a board balanced
on a log.

The mean is the sum of the
scores divided by the number
of scores.

S sum of; add up all the scores follow-
ing this symbol.

X scores in the distribution of the
variable X.

T I P F O R S U C C E S S
Think of each formula as a statisti-
cal recipe, with statistical symbols
as ingredients. Before you use
each formula, be sure you know
what each symbol stands for. Then
carefully follow the formula to
come up with the end result.

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36 Chapter 2

N stands for number—the number of scores in a distribution. In our example,
there are 10 scores. Thus, N equals 10.1

Overall, the formula says to divide the sum of all the scores in the distribution of
the variable X by the total number of scores, N. In the dreams example, this means
you divide 60 by 10. Put in terms of the formula,

Additional Examples of Figuring the Mean
Consider the examples from Chapter 1. The stress ratings of the 30 students in the
first week of their statistics class (based on Aron et al., 1995) were:

8, 7, 4, 10, 8, 6, 8, 9, 9, 7, 3, 7, 6, 5, 0, 9, 10, 7, 7, 3, 6, 7, 5, 2, 1, 6, 7, 10, 8, 8

In Chapter 1 we summarized all these numbers into a frequency table (Table 1–3).
You can now summarize all this information as a single number by figuring the
mean. Figure the mean by adding up all the stress ratings and dividing by the num-
ber of stress ratings. That is, you add up the 30 stress ratings:

, for a total of 193. Then you divide this
total by the number of scores, 30. In terms of the formula,

This tells you that the average stress rating was 6.43 (after rounding off). This is
clearly higher than the middle of the 0–10 scale. You can also see this on a graph.
Think again of the histogram as a pile of blocks on a board and the mean of 6.43 as
the point where the board balances on the fulcrum (see Figure 2–3). This single rep-
resentative value simplifies the information in the 30 stress scores.

M =
gX

N
=

193

30
= 6.43

7 + 5 + 2 + 1 + 6 + 7 + 10 + 8 + 8
8 + 6 + 8 + 9 + 9 + 7 + 3 + 7 + 6 + 5 + 0 + 9 + 10 + 7 + 7 + 3 + 6 +

8 + 7 + 4 + 10 +

M =
gX
N
=
60

= 6

N number of scores in a distribution.

T I P F O R S U C C E S S
When an answer is not a whole
number, we suggest that you use
two more decimal places in the an-
swer than for the original numbers.
In this example, the original num-
bers did not use decimals, so we
rounded the answer to two deci-
mal places.

7
6
5
4
3
2
1

0
0 1 2 3 4 5 6 7 8 9 10

Balance Point

F
re

qu
en

6.43Stress Rating

Figure 2–3 Analogy of blocks on a board balanced on a fulcrum showing the means
for 30 statistics students’ ratings of their stress level. (Data based on Aron et al., 1995.)

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Central Tendency and Variability 37

Similarly, consider the Chapter 1 example of students’ social interactions
(McLaughlin-Volpe et al., 2001). The actual number of interactions over a week for
the 94 students are listed on page 8. In Chapter 1, we organized the original scores
into a frequency table (see Table 1–5). We can now take those same 94 scores, add
them up, and divide by 94 to figure the mean:

This tells us that during this week these students had an average of 17.39 social in-
teractions. Figure 2–4 shows the mean of 17.39 as the balance point for the 94 social
interaction scores.

Steps for Figuring the Mean
Figure the mean in two steps.

❶ Add up all the scores. That is, figure ΣX.
❷ Divide this sum by the number of scores. That is, divide ΣX by N.

The Mode
The mode is another measure of central tendency. The mode is the most common
single value in a distribution. In our dreams example, the mode is 8. This is because
there are three students with 8 dreams and no other number of dreams with as many
students. Another way to think of the mode is that it is the value with the largest
frequency in a frequency table, the high point or peak of a distribution’s histogram
(as shown in Figure 2–5).

M =
gX
N
=

1,635

94
=

17.39

10
9
8
7
6
5
4
3
2
1

0
2.5 7.5 12.5 22.5 27.5 42.5 47.5

Number of Social Interactions
in a Week

F
re
qu
en
cy

32.5 37.517.5

Balance Point
17.39

Figure 2–4 Analogy of blocks on a board balanced on a fulcrum illustrating the
mean for number of social interactions during a week for 94 college students. (Data from
McLaughlin-Volpe et al., 2001.)

mode value with the greatest frequency
in a distribution.

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38 Chapter 2

In a perfectly symmetrical unimodal distribution, the mode is the same as the
mean. However, what happens when the mean and the mode are not the same? In
that situation, the mode is usually not a very good way of describing the central ten-
dency of the scores in the distribution. In fact, sometimes researchers compare the
mode to the mean to show that the distribution is not perfectly symmetrical. Also, the
mode can be a particularly poor representative value because it does not reflect many
aspects of the distribution. For example, you can change some of the scores in a
distribution without affecting the mode—but this is not true of the mean, which is
affected by any change in the distribution (see Figure 2–6).

5 6 7 8 91 2 3 4

Mode = 8

Figure 2–5 Mode as the high point in a distribution’s histogram, using the example of
the number of dreams during a week for 10 students.

5 6 7 8 9 102 3 4

5 6 7 8 9 102 3 4
5 6 7 8 9 102 3 4

Mean = 8.30

Mode = 8

Mean = 5.10

Mode = 8
Mode = 8

Mean = 7

Figure 2–6 Effect on the mean and on the mode of changing some scores, using the
example of the number of dreams during a week for 10 students.

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Central Tendency and Variability 39

7 7633 98881

Median

Figure 2–7 The median is the middle score when scores are lined up from lowest to
highest, using the example of the number of dreams during a week for 10 students.

T I P F O R S U C C E S S
When figuring the median, remem-
ber that the first step is to line up
the scores from lowest to highest.
Forgetting to do this is the most
common mistake students make
when figuring the median.

median middle score when all the
scores in a distribution are arranged from
lowest to highest.

outlier score with an extreme value
(very high or very low) in relation to the
other scores in the distribution.

On the other hand, the mode is the usual way of describing the central tendency
for a nominal variable. For example, if you know the religions of a particular group of
people, the mode tells you which religion is the most frequent. However, when it
comes to the numerical variables that are most common in psychology research, the
mode is rarely used.

The Median
Another alternative to the mean is the median. If you line up all the scores from low-
est to highest, the middle score is the median. Figure 2–7 shows the scores for the
number of dreams lined up from lowest to highest. In this example, the fifth and
sixth scores (the two middle ones) are both 7s. Either way, the median is 7.

When you have an even number of scores, the median is between two different
scores. In that situation, the median is the average (the mean) of the two scores.

Steps for Finding the Median
Finding the median takes three steps.

❶ Line up all the scores from lowest to highest.
❷ Figure how many scores there are to the middle score by adding 1 to the num-

ber of scores and dividing by 2. For example, with 29 scores, adding 1 and divid-
ing by 2 gives you 15. The 15th score is the middle score. If there are 50 scores,
adding 1 and dividing by 2 gives you 251⁄2. Because there are no half scores, the
25th and 26th scores (the scores on either side of 251⁄2) are the middle scores.

❸ Count up to the middle score or scores. If you have one middle score, this is
the median. If you have two middle scores, the median is the average (the mean)
of these two scores.

Comparing the Mean, Mode, and Median
Sometimes, the median is better than the mean (and mode) as a representative value for
a group of scores. This happens when a few extreme scores would strongly affect the
mean but would not affect the median. Reaction time scores are a common example in
psychology research. Suppose you are asked to press a key as quickly as possible when
a green circle is shown on the computer screen. On five showings of the green circle,
your times (in seconds) to respond are .74, .86, 2.32, .79, and .81. The mean of these
five scores is 1.1040: that is, . However, this mean is
very much influenced by the one very long time (2.32 seconds). (Perhaps you were dis-
tracted just when the green circle was shown.) The median is much less affected by the
extreme score. The median of these five scores is .81—a value that is much more rep-
resentative of most of the scores. Thus, using the median deemphasizes the one ex-
treme time, which is probably appropriate. An extreme score like this is called an
outlier. In this example, the outlier was much higher than the other scores, but in other
cases an outlier may be much lower than the other scores in the distribution.

(©X)>N = 5.52> 5 = 1.1040

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40 Chapter 2

The importance of whether you use the mean, mode, or median can be seen in a
controversy among psychologists studying the evolutionary basis of human mate
choice. One set of theorists (e.g., Buss & Schmitt, 1993) argue that over their lives, men
should prefer to have many partners, but women should prefer to have just one reliable
partner. This is because a woman can have only a small number of children in a lifetime
and her genes are most likely to survive if those few children are well taken care of. Men,
however, can have a great many children in a lifetime. Therefore, according to the theory,
a shotgun approach is best for men, because their genes are most likely to survive if they
have a great many partners. Consistent with this assumption, evolutionary psycholo-
gists have found that men report wanting far more partners than do women.

Other theorists (e.g., Miller & Fishkin, 1997), however, have questioned this view.
They argue that women and men should prefer about the same number of partners.
This is because individuals with a basic predisposition to seek a strong intimate bond
are most likely to survive infancy. This desire for strong bonds, they argue, remains in
adulthood. These theorists also asked women and men how many partners they wanted.
They found the same result as the previous researchers when using the mean: men
wanted an average of 64.3 partners, women an average of 2.8 partners. However, the
picture looks drastically different if you look at the median or mode (see Table 2–1).
Figure 2–8, taken directly from their article, shows why. Most women and most men
want just one partner. A few want more, some many more. The big difference is that

P
er

ce
nt

ag
e

of
M

en
a

nd
W

m
en

–
–
–
–

–

–
–
–
–

50 –
–
–
–
–

40 –
–
–
–
–

30 –
–
–
–
–

20 –
–
–
–
–

10 –
–
–
–
–

0 – – 0 – 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10
–

11
–2

0 –
21

–3
0 –

31
–4

0 –
41

–5
0 –

51
–6

0 –
61

–7
0 –

71
–8

0 –
81

–9
0 –

91
–1

00
–

10
0–

10
00

–
10

01
–1

00
00

–

Women %

Men %

Number of Partners Desired in the Next 30 Years

Women 2.8 1

Men 64.3 1

Mean Median

Measures of central tendency

Figure 2–8 Distributions for men and women for the ideal number of partners desired
over 30 years. Note: To include all the data, we collapsed across categories farther out on the tail of
these distributions. If every category represented a single number, it would be more apparent that
the tail is very flat and that distributions are even more skewed than is apparent here.

Source: Miller, L. C., & Fishkin, S. A. (1997). On the dynamics of human bonding and reproductive suc-
cess: Seeking windows on the adapted-for-human-environmental interface. In J. Simpson & D. T. Kenrick
(Eds.), Evolutionary social psychology (pp. 197–235). Mahwah, NJ: Erlbaum.

Table 2–1 Responses of 106
Men and 160 Women to the Question,
“How many partners would you ideally
desire in the next 30 years?”

Mean Median Mode

Women 2.8 1 1

Men 64.3 1 1

Source: Data from Miller & Fishkin (1997).

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Central Tendency and Variability 41

there are a lot more men in the small group that want many more than one partner.
These results were also replicated in a more recent study (Pedersen et al., 2002).

So which theory is right? You could argue either way from these results. The point
is that focusing just on the mean can clearly misrepresent the reality of the distribution.
As this example shows, the median is most likely to be used when a few extreme
scores would make the mean unrepresentative of the main body of scores. Figure 2–9
illustrates this point, by showing the relative location of the mean, mode, and median
for three types of distribution that you learned about in Chapter 1. The distribution in
Figure 2–9a is skewed to the left (negatively skewed); the long tail of the distribution
points to the left. The mode in this distribution is the highest point of the distribution,
which is on the far right hand side of the distribution. The median is the point at which
half of the scores are above that point and half are below. As you can see, for that to
happen, the median must be a lower value than the mode. Finally, the mean is strongly
influenced by the very low scores in the long tail of the distribution and is thus a lower
value than the median. Figure 2–9b shows the location of the mean, mode, and median
for a distribution that is skewed to the right (positively skewed). In this case, the mean
is a higher value than either the mode or median because the mean is strongly influ-
enced by the very high scores in the long tail of the distribution. Again, the mode is the
highest point of the distribution, and the median is between the mode and the mean.
In Figures 2–9a and 2–9b, the mean is not a good representative value of the scores,
because it is unduly influenced by the extreme scores.

Mean Median Mode

MeanMedianMode

Mean
Mode

Median

(a)

(b)

(c)

Figure 2–9 Locations of the mean, mode, and median on (a) a distribution skewed to
the left, (b) a distribution skewed to the right, and (c) a normal curve.

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42 Chapter 2

Figure 2–9c shows a normal curve. As for any distribution, the mode is the high-
est point in the distribution. For a normal curve, the highest point falls exactly at the
midpoint of the distribution. This midpoint is the median value, since half of the
scores in the distribution are below that point and half are above it. The mean also
falls at the same point because the normal curve is symmetrical about the midpoint,
and every score in the left hand side of the curve has a matching score on the right
hand side. So, for a normal curve, the mean, mode, and median are always the same
value.

In some situations psychologists use the median as part of more complex statis-
tical methods. Also, the median is the usual way of describing the central tendency
for a rank-order variable. Otherwise, unless there are extreme scores, psychologis

almost always use the mean as the representative value of a group of scores. In fact,
as you will learn, the mean is a fundamental building block for most other statistical
techniques.

A summary of the mean, mode, and median as measures of central tendency is
shown in Table 2–2.

Table 2–2 Summary of Measures of Central Tendency

Measure Definition When Used

How are you doing?

1. Name and define three measures of central tendency.
2. Write the formula for the mean and define each of the symbols.
3. Figure the mean of the following scores: 2, 8, 3, 6, and 6.
4. For the following scores find (a) the mean, (b) the mode, and (c) the median: 5,

3, 2, 13, 2. (d) Why is the mean different from the median?

Answers

1.The mean is the ordinary average, the sum of the scores divided by the num-
ber of scores. The mode is the most frequent score in a distribution. The me-
dian is the middle score; that is, if you line the scores up from lowest to
highest, it is the halfway score.

2.The formula for the mean is is the mean; is the symbol for
“sum of”—add up all the scores that follow; Xis the variable whose scores
you are adding up; Nis the number of scores.

3..
4.(a) The mean is 5; (b) the mode is 2; (c) the median is 3; (d) The mean is differ-

ent from the median because the extreme score (13) makes the mean higher
than the median.

M=(©X)>N=(2+8+3+6+6)>5=5

g M=(©X)>N. M

Mean Sum of the scores divided by the
number of scores

• With equal-interval variables
• Very commonly used in psychology

research

Mode Value with the greatest frequency
in a distribution

• With nominal variables
• Rarely used in psychology research

Median Middle score when all the scores in
a distribution are arranged from
lowest to highest

• With rank-ordered variables
• When a distribution has one or more

outliers
• Rarely used in psychology research

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Central Tendency and Variability 43

(a) (b)

1.70
Mean

3.20

Mean
^

3.20
Mean

^
3.20
Mean

^
2.50

Mean
^
3.20
Mean
^

Figure 2–10 Examples of distributions with (a) the same mean but different amounts
of spread, and (b) different means but the same amount of spread.

Variability
Researchers also want to know how spread out the scores are in a distribution. This
shows the amount of variability in the distribution. For example, suppose you were
asked, “How old are the students in your statistics class?” At a city-based university
with many returning and part-time students, the mean age might be 29. You could
answer, “The average age of the students in my class is 29.” However, this would not
tell the whole story. You could have a mean of 29 because every student in the class
was exactly 29 years old. If this is the case, the scores in the distribution are not
spread out at all. In other words, there is no variation, or variability, among the
scores. You could also have a mean of 29 because exactly half the class members
were 19 and the other half 39. In this situation, the distribution is much more spread
out; there is considerable variability among the scores in the distribution.

You can think of the variability of a distribution as the amount of spread of the
scores around the mean. Distributions with the same mean can have very different
amounts of spread around the mean; Figure 2–10a shows histograms for three differ-
ent frequency distributions with the same mean but different amounts of spread
around the mean. A real-life example of this is shown in Figure 2–11, which shows
the distributions of the housing prices in two neighborhoods: one with diverse hous-
ing types and the other with a consistent type of housing. As with Figure 2–10a, the
mean housing price is the same in each neighborhood. However, the distribution for

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44 Chapter 2

MeanHousing
Prices

Neighborhood with
Consistent
Type of Housing

MeanHousing
Prices

Neighborhood with
Diverse
Types of Housing

Figure 2–11 Example of two distributions with the same mean but different amounts
of spread: housing prices for a neighborhood with diverse types of housing and for a neigh-
borhood with a consistent type of housing.

variance measure of how spread out a
set of scores are; average of the squared
deviations from the mean.

deviation score score minus the
mean.

squared deviation score square of
the difference between a score and the
mean.

the neighborhood with diverse housing types is much more spread out around the
mean than the distribution for the neighborhood that has a consistent type of housing.
This tells you that there is much greater variability in the prices of housing in the
neighborhood with diverse types of housing than in the neighborhood with a consis-
tent housing type. Also, distributions with different means can have the same amount
of spread around the mean. Figure 2–10b shows three different distributions with dif-
ferent means but the same amount of spread. So, while the mean provides a represen-
tative value of a group of scores, it doesn’t tell you about the variability of the scores.
You will now learn about two measures of the variability of a group of scores: the
variance and standard deviation.2

The Variance
The variance of a group of scores tells you how spread out the scores are around the
mean. To be precise, the variance is the average of each score’s squared difference
from the mean.

Here are the four steps to figure the variance:

❶ Subtract the mean from each score. This gives each score’s deviation score,
which is how far away the score is from the mean.

❷ Square each of these deviation scores (multiply each by itself). This gives each
score’s squared deviation score.

❸ Add up the squared deviation scores. This total is called the sum of squared
deviations.

❹ Divide the sum of squared deviations by the number of scores. This gives the
average (the mean) of the squared deviations, called the variance.

Suppose one distribution is more spread out than another. The more spread-out
distribution has a larger variance because being spread out makes the deviation
scores bigger. If the deviation scores are bigger, the squared deviation scores and the
average of the squared deviation scores (the variance) are also bigger.

sum of squared deviations total of
all the scores of each score’s squared dif-
ference from the mean.

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Central Tendency and Variability 45

In the example of the class in which everyone was exactly 29 years old, the
variance would be exactly 0. That is, there would be no variance (which makes
sense, because there is no variability among the ages). (In terms of the numbers,
each person’s deviation score would be ; 0 squared is 0. The average
of a bunch of zeros is 0.) By contrast, the class of half 19-year-olds and half 39-
year-olds would have a rather large variance of 100. (The 19-year-olds would each
have deviation scores of . The 39-year-olds would have deviation
scores of . All the squared deviation scores, which are either
squared or 10 squared, come out to 100. The average of all 100s is 100.)

The variance is extremely important in many statistical procedures you will learn
about later. However, the variance is rarely used as a descriptive statistic. This is be-
cause the variance is based on squared deviation scores, which do not give a very
easy-to-understand sense of how spread out the actual, nonsquared scores are. For ex-
ample, a class with a variance of 400 clearly has a more spread-out distribution than
one whose variance is 10. However, the number 400 does not give an obvious insight
into the actual variation among the ages, none of which is anywhere near 400.3

The Standard Deviation
The most widely used way of describing the spread of a group of scores is the
standard deviation. The standard deviation is directly related to the variance and is
figured by taking the square root of the variance. There are two steps in figuring the
standard deviation.

❶ Figure the variance.
❷ Take the square root. The standard deviation is the positive square root of the

variance. (Any number has both a positive and a negative square root. For ex-
ample, the square root of 9 is both and .)

If the variance of a distribution is 400, the standard deviation is 20. If the vari-
ance is 9, the standard deviation is 3.

The variance is about squared deviations from the mean. Therefore, its square
root, the standard deviation, is about direct, ordinary, not-squared deviations from the
mean. Roughly speaking, the standard deviation is the average amount that scores
differ from the mean. For example, consider a class where the ages have a standard
deviation of 20 years. This tells you that the ages are spread out, on the average, about
20 years in each direction from the mean. Knowing the standard deviation gives you
a general sense of the degree of spread.4

The standard deviation does not, however, perfectly describe the shape of the
distribution. For example, suppose the distribution of the number of children in fam-
ilies in a particular country has a mean of 4 and standard deviation of 1. Figure 2–12
shows several possibilities of the distribution of number of children, all with a mean
of 4 and a standard deviation of 1.

Formulas for the Variance and the Standard Deviation
We have seen that the variance is the average squared deviation from the mean. Here
is the formula for the variance.

(2–2)SD2 =
g(X – M)2

– 3+ 3

– 1039 – 29 = 10
19 – 29 = – 10

29 – 29 = 0

standard deviation square root of the
average of the squared deviations from
the mean; the most common descriptive
statistic for variation; approximately the
average amount that scores in a distribu-
tion vary from the mean.

The variance is the sum of the
squared deviations of the
scores from the mean, divided
by the number of scores.

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46 Chapter 2

SD2 variance.

SD standard deviation.
SD2 is the symbol for the variance. This may seem surprising. SD is short for

standard deviation. The symbol SD2 emphasizes that the variance is the standard devi-
ation squared. (Later, you will learn other symbols for the variance, S 2 and —the
lowercase Greek letter sigma squared. The different symbols are for different situa-
tions in which the variance is used. In some cases, it is figured slightly differently.)

The top part of the formula is the sum of squared deviations. X is for each score
and M is the mean. Thus, X – M is the score minus the mean, the deviation score. The
superscript number (2) tells you to square each deviation score. Finally, the sum sign
(Σ) tells you to add up all these squared deviation scores.

The sum of squared deviations of the scores from the mean, which is called the
sum of squares for short, has its own symbol, SS. Thus, the variance formula can be
written using SS instead of Σ(X – M)2:

(2–3)

Whether you use the simplified symbol SS or the full description of the sum of
squared deviations, the bottom part of the formula is just N, the number of scores.

SD2 =
SS

�2sum of squares (SS) sum of squared
deviations.

T I P F O R S U C C E S S
The sum of squared deviations is
an important part of many of the
procedures you learn in later chap-
ters; so be sure you fully under-
stand it, as well as how it is
figured.

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

Figure 2–12 Some possible distributions for family size in a country where the mean
is 4 and the standard deviation is 1.

The variance is the sum of
squares divided by the
number of scores. ISB

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Central Tendency and Variability 47

That is, the formula says to divide the sum of the squared deviation scores by the
number of scores in the distribution.

The standard deviation is the square root of the variance. So, if you already
know the variance, the formula is

(2–4)

The formula for the standard deviation, starting from scratch, is the square root
of what you figure for the variance:

(2–5)

(2–6)

Examples of Figuring the Variance and Standard Deviation
Table 2–3 shows the figuring for the variance and standard deviation for the number
of dreams example. (The table assumes you have already figured the mean to be 6
dreams.) Usually, it is easiest to do your figuring using a calculator, especially one
with a square root key. The standard deviation of 2.57 tells you that roughly speak-
ing, on the average, the number of dreams vary by about 21⁄2 from the mean of 6.

Table 2–4 shows the figuring for the variance and standard deviation for the
example of students’ number of social interactions during a week (McLaughlin-
Volpe et al., 2001). (To save space, the table shows only the first few and last few
scores.) Roughly speaking, this result tells you that a student’s number of social in-
teractions in a week varies from the mean (of 17.39) by an average of 11.49. This can
also be shown on a histogram (see Figure 2–13).

Measures of variability, such as the variance and standard deviation, are heavily
influenced by the presence of one or more outliers (extreme values) in a distribution.

SD = A
SS

SD = B
©(X – M)2

SD = 2SD2

Table 2–3 Figuring the Variance and Standard Deviation in the Number of Dreams Example

Score

(Number of

Dreams)
�

Mean Score
(Mean Number

of Dreams)
� Deviation

Score

Squared
Deviation

Score

7 6 1 1

8 6 2 4

8 6 2 4
7 6 1 1

3 6 9

1 6 25

6 6 0 0

9 6 3 9

3 6 9
8 6 2 4

Standard deviation = SD = 2SD 2 = 26.60 = 2.57
Variance = SD 2 =

g(X – M )2

N
=

SS
N

=
66
10

= 6.60

©: 0

– 3

– 5
– 3

The standard deviation is the
square root of the variance.

The standard deviation is the
square root of the result of
taking the sum of the squared
deviations of the scores from
the mean divided by the
number of scores.

The standard deviation is the
square root of the result of
taking the sum of squares
divided by the number of
scores.

T I P F O R S U C C E S S
When figuring the variance and
standard deviation, lay your work
out as in Tables 2–3 and 2–4. This
helps you follow all the steps and
end up with the correct answers.

T I P F O R S U C C E S S
Always check that your answers
make intuitive sense. For example,
looking at the scores for the dreams
example, a standard deviation—
which, roughly speaking, repre-
sents the average amount that the
scores vary from the mean—of
2.57 makes sense. If your answer
had been 21.23, however, it would
mean that, on average, the number
of dreams varied by more than 20
from the mean of 6. Looking at the
group of scores, that just couldn’t
be true.

T I P F O R S U C C E S S
Notice in Table 2–3 that the devia-
tion scores (shown in the third col-
umn) add up to 0. The sum of the
deviation scores is always 0 (or
very close to 0, allowing for round-
ing error). So, to check your figur-
ing, always sum the deviation
scores. If they do not add up to 0,
do your figuring again!

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48 Chapter 2

Table 2–4 Figuring the Variance and Standard Deviation for Number of Social Interactions
During a Week for 94 College Students

Number of
Interactions �

Mean Number
of Interactions �

Deviation
Score

Squared
Deviation

Score

48 17.39 30.61 936.97

15 17.39 5.71

33 17.39 15.61 243.67

3 17.39 207.07

21 17.39 3.61 13.03

– – – –

– – – –
– – – –

35 17.39 17.61 310.11
9 17.39 70.39

30 17.39 12.61 159.01

8 17.39 88.17

26 17.39 8.61 74.13

12,406.44

Source: Data from McLaughlin-Volpe et al. (2001).

Standard deviation = 2SD 2 = 2131.98 = 11.49
Variance = SD 2 =

g(X – M )2

N
=

12,406.44
94

= 131.98

©: 0.00
– 9.39

– 8.39

– 14.39

– 2.39

16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1

2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5

Number of Social Interactions in a Week

F
re
qu
en
cy

^ ^ ^ ^ ^ ^ ^ ^ ^ ^

1 SD

FREQUENCY
12
16
16
16
10
11
4

3
3

INTERVAL
0 – 4
5 – 9

10 – 14
15 – 19
20 – 24
25 – 29
30 – 34
35 – 39
40 – 44
45 – 49

1 SD

Figure 2–13 The standard deviation as the distance along the base of a histogram,
using the example of number of social interactions in a week. (Data from McLaughlin-Volpe
et al., 2001.)

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Central Tendency and Variability 49

T I P F O R S U C C E S S
A common mistake when figuring
the standard deviation is to jump
straight from the sum of squared
deviations to the standard devia-
tion (by taking the square root of
the sum of squared deviations).
Remember, before finding the
standard deviation, first figure the
variance (by dividing the sum of
squared deviations by the number
of scores, N). Then take the square
root of the variance to find the
standard deviation.

The scores in the number of dreams example were 7, 8, 8, 7, 3, 1, 6, 9, 3, 8, and we
figured the standard deviation of the scores to be 2.57. Now imagine that one addi-
tional person is added to the study and that the person reports having 21 dreams in the
past week. The standard deviation of the scores would now be 4.96, which is almost
double the size of the standard deviation without this additional single score.

Computational and Definitional Formulas
In actual research situations, psychologists must often figure the variance and the
standard deviation for distributions with many scores, often involving decimals or
large numbers. In the days before computers, this could make the whole process
quite time-consuming, even with a calculator. To deal with this problem, in the old
days researchers developed various shortcuts to simplify the figuring. A shortcut for-
mula of this type is called a computational formula.

The traditional computational formula for the variance of the kind we are dis-
cussing in this chapter is as follows:

(2–7)

means that you square each score and then take the sum of the squared
scores. However, means that you first add up all the scores and then take the
square of this sum. Although this sounds complicated, this formula was actually eas-
ier to use than the one you learned before if a researcher was figuring the variance
for a lot of numbers by hand or even with an old-fashioned handheld calculator, be-
cause the researcher did not have to first find the deviation score for each score.

However, these days computational formulas are mainly of historical interest.
They are used by researchers only on rare occasions when computers with statistics
software are not readily available to do the figuring. In fact, today, even many hand-
held calculators are set up so that you need only enter the scores and press a button
or two to get the variance and the standard deviation.

In this book we give a few computational formulas just so that you have them if
you someday do a research project with a lot of numbers and you don’t have access to
statistical software. However, we very definitely recommend not using the computa-
tional formulas when you are learning statistics, even if they might save you a few min-
utes of figuring a practice problem. The computational formulas usually make it much
harder to understand the meaning of what you are figuring. The only reason for figuring
problems at all by hand when you are learning statistics is to reinforce the underlying
principles. Thus, you would be undermining the whole point of the practice problems if
you use a formula that had a complex relation to the basic logic. The formulas we give
you for the practice problems and for all the examples in the book are designed to help
strengthen your understanding of what the figuring means. Thus, the usual formula we
give for each procedure is what statisticians call a definitional formula.

The Importance of Variability in Psychology Research
Variability is an important topic in psychology research because much of the re-
search focuses on explaining variability. We will use a couple of examples to show
what we mean by “explaining variability.” As you might imagine, different students
experience different levels of stress with regard to learning statistics: Some experi-
ence little stress; for other students, learning statistics can be a source of great stress.
So, in this example, explaining variability means identifying the factors that explain
why students differ in the amount of stress they experience. Perhaps how much
experience students have had with math explains some of the variability. That is,

(©X)2
©X2

SD2 =
gX2 – A AgX B 2 > N B

computational formula equation
mathematically equivalent to the defini-
tional formula. Easier to use for figuring
by hand, it does not directly show the
meaning of the procedure.

definitional formula equation for a
statistical procedure directly showing
the meaning of the procedure.

The variance is the sum of the
squared scores minus the
result of taking the sum of all
the scores, squaring this sum
and dividing by the number of
scores, then taking this whole
difference and dividing it by
the number of scores.

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50 Chapter 2

according to this explanation, the differences (the variability) among students in
amount of stress are partially due to the differences (the variability) among students
in the amount of experience they have had with math. Thus, the variation in math
experience partially explains, or accounts for, the variation in stress. What factors
might explain the variation in students’ number of weekly social interactions? Per-
haps a factor is variation in the extraversion of students, with more extraverted stu-
dents tending to have more interactions. Or perhaps it is variation in gender, with one
gender having consistently more interactions than the other. Much of the rest of this
book focuses on procedures for evaluating and testing whether variation in some
specific factor (or factors) explains the variability in some variable of interest.

The Variance as the Sum of Squared Deviations
Divided by
Researchers often use a slightly different kind of variance. We have defined the vari-
ance as the average of the squared deviation scores. Using that definition, you divide
the sum of the squared deviation scores by the number of scores (that is, the variance is
SS�N). But you will learn in Chapter 7 that for many purposes it is better to define the
variance as the sum of squared deviation scores divided by 1 less than the number of
scores. In other words, for those purposes the variance is the sum of squared deviations
divided by (that is, variance is SS�[ ]). (As you will learn in Chapter 7,
you use this dividing by approach when you have scores from a particular group
of people and you want to estimate what the variance would be for the larger group of
people whom these individuals represent.)

The variances and standard deviations given in research articles are usually fig-
ured using SS�( ). Also, when calculators or computers give the variance or
the standard deviation automatically, they are usually figured in this way (for exam-
ple, see the Using SPSS section at the end of this chapter). But don’t worry. The ap-
proach you are learning in this chapter of dividing by N (that is, figuring variance as
SS�N) is entirely correct for our purpose here, which is to use descriptive statistics to
describe the variation in a particular group of scores. It is also entirely correct for the
material you learn in Chapters 3 through 6. We mention this other approach (vari-
ance as SS�[ ]) now only so that you will not be confused if you read about
variance or standard deviation in other places or if your calculator or a computer pro-
gram gives a surprising result. To keep things simple, we wait to discuss the dividing
by approach until it is needed, starting in Chapter 7.

– 1

N – 1

N – 1
N – 1N – 1

N � 1

How are you doing?

1. (a) Define the variance and (b) indicate what it tells you about a distribution
and how this is different from what the mean tells you.

2. (a) Define the standard deviation; (b) describe its relation to the variance; and
(c) explain what it tells you approximately about a group of scores.

3. Give the full formula for the variance and indicate what each of the symbols
means.

4. Figure the (a) variance and (b) standard deviation for the following scores: 2,
4, 3, and 7 ( ).

5. Explain the difference between a definitional and a computational formula.
6. What is the difference between the formula for the variance you learned in this

chapter and the formula that is typically used to figure the variance in re-
search articles?

M = 4

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Central Tendency and Variability 51

Answers

1.(a) The variance is the average of the squared deviation of each score from
the mean. (b) The variance tells you about how spread out the scores are
(that is, their variability), while the mean tells you the central tendency of the
distribution.

2.(a) The standard deviation is the square root of the average of the squared de-
viations from the mean. (b) The standard deviation is the square root of the
variance. (c) The standard deviation tells you approximately the average
amount that scores differ from the mean.

3.is the variance.means the sum of what
follows. Xis for the scores for the variable being studied. Mis the mean of the
scores. Nis the number of scores.

4. (a)Variance:

(b) Standard deviation:
5.A definitional formula is the standard formula in the straightforward form that

shows the meaning of what the formula is figuring. A computational formula is
a mathematically equivalent variation of the definitional formula, but the com-
putational formula tends not to show the underlying meaning. Computational
formulas were often used before computers were available and researchers
had to do their figuring by hand with a lot of scores.

6.The formula for the variance in this chapter divides the sum of squares by the
number of scores (that is, SS�N). The variance in research articles is usually
figured by dividing the sum of squares by one less than the number of scores
(that is, SS�[]). N-1

SD=2SD
2

=23.50=1.87.
14>4=3.50. (7-4)

2
]>4=

SD
2
=[©(X-M)

2
]>N=[(2-4)

2
+(4-4)

2
+(3-4)

2
+

=[©1X-M2
2
]>N. SD

You are learning statistics for the fun of it, right? No? Or
maybe so, after all. If you become a psychologist, at some
time or other you will form a hypothesis, gather data, and
analyze them. (Even if you plan a career as a psychother-
apist or other mental health practitioner, you will proba-
bly eventually wish to test an idea about the nature of
your patients and their difficulties.) That hypothesis—
your own original idea—and the data you gather to test
it are going to be very important to you. Your heart may
well be pounding with excitement as you analyze the
data.

Consider some of the comments of social psycholo-
gists we interviewed for our book The Heart of Social
Psychology (Aron & Aron, 1989). Deborah Richardson,
who studies interpersonal relationships, confided that her
favorite part of being a social psychologist is looking at
the statistical output of the computer analyses:

It’s like putting together a puzzle. . . . It’s a highly arous-
ing, positive experience for me. I often go through periods

of euphoria. Even when the data don’t do what I want
them to do . . . [there’s a] physiological response. . . .
It’s exciting to see the numbers come off—Is it actually
the way I thought it would be?—then thinking about the
alternatives.

Harry Reis, former editor of the Journal of Personality
and Social Psychology, sees his profession the same
way:

By far the most rewarding part is when you get a new
data set and start analyzing it and things pop out, partly
a confirmation of what led you into the study in the first
place, but then also other things. . . . “Why is that?”
Trying to make sense of it. The kind of ideas that come
from data. . . . I love analyzing data.

Bibb Latane, an eminent psychologist known for,
among other things, his work on why people don’t al-
ways intervene to help others who are in trouble, reports
eagerly awaiting

B O X 2 – 1 The Sheer Joy (Yes, Joy) of Statistical Analysis

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52 Chapter 2

Controversy: The Tyranny of the Mean
Looking in the behavioral and social science research journals, you would think that
statistical methods are their sole tool and language, but there have also been rebel-
lions against the reign of statistics. We are most familiar with this issue in psychology,
where the most unexpected opposition came from the leader of behaviorism, the
school of psychology most dedicated to keeping the field strictly scientific.

Behaviorism opposed the study of inner states because inner events are impossi-
ble to observe objectively. (Today most research psychologists claim to measure inner
events indirectly but objectively.) Behaviorism’s most famous advocate, B. F. Skinner,
was quite opposed to statistics. Skinner even said, “I would much rather see a graduate
student in psychology taking a course in physical chemistry than in statistics. And I
would include [before statistics] other sciences, even poetry, music, and art” (Evans,
1976, p. 93).

Skinner was constantly pointing to the information lost by averaging the results
of a number of cases. For instance, Skinner (1956) cited the example of three
overeating mice—one naturally obese, one poisoned with gold, and one whose hy-
pothalamus had been altered. Each had a different curve for learning to press a bar
for food. If these learning curves had been summed or merged statistically, the result
would have represented no actual eating habits of any real mouse. As Skinner said,
“These three individual curves contain more information than could probably ever
be generated with measures requiring statistical treatment, yet they will be viewed
with suspicion by many psychologists because they are single cases” (p. 232).

In clinical psychology and the study of personality, voices have always been
raised in favor of the in-depth study of one person instead of or as well as the aver-
aging of persons. The philosophical underpinnings of the in-depth study of individ-
uals can be found in phenomenology, which began in Europe after World War I
(Husserl, 1970). This viewpoint has been important throughout the social sciences,
not just in psychology.

Today, the rebellion in psychology is led by qualitative research methodologists
(e.g., McCracken, 1988), an approach that is much more prominent in other behav-
ioral and social sciences. The qualitative research methods, developed mainly in an-
thropology, can involve long interviews or observations of a few individuals. The
highly skilled researcher decides, as the event is taking place, what is important to
remember, record, and pursue through more questions or observations. The mind of
the researcher is the main tool because, according to this approach, only that mind
can find the important relationships among the many categories of events arising in
the respondent’s speech.

Many who favor qualitative methods argue for a blend: First, discover the im-
portant categories through a qualitative approach. Then, determine their incidence in

. . . the first glimmerings of what came out . . . [and]
using them to shape what the next question should be . . .
You need to use everything you’ve got, . . . every bit of
your experience and intuition. It’s where you have the
biggest effect, it’s the least routine. You’re in the room
with the tiger, face to face with the core of what you are
doing, at the moment of truth.

Bill Graziano, whose work integrates developmental,
personality, and social psychology, calls the analysis of
his data “great fun, just great fun.” And in the same vein,
Margaret Clark, who studies emotion and cognition, de-
clares that “the most fun of all is getting the data and
looking at them.”

So you see? Statistics in the service of your own cre-
ative ideas can be a pleasure indeed.

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Central Tendency and Variability 53

the larger population through quantitative methods. Too often, these advocates
argue, quantitative researchers jump to conclusions about a phenomenon without
first exploring the human experience of it through free-response interviews or
observations.

Finally, Carl Jung, founder of Jungian psychology, sometimes spoke of the “sta-
tistical mood” and its effect on a person’s feeling of uniqueness. Jung had no prob-
lem with statistics—he used them in his own research. He was concerned about the
cultural impact of this “statistical mood”—much like the impact of being on a
jammed subway and observing the hundreds of blank faces and feeling diminished,
“one of a crowd.” He held that the important contributions to culture tend to come
from people who feel unique and not ordinary. As we increasingly describe our-
selves statistically—“90% of men under thirty think . . .”—we tend to do just that,
think like 90% of men under thirty. To counteract this mood, Jungian analyst Marie
Louise von Franz (1979) wrote, “An act of loyalty is required towards one’s own
feelings” (pp. IV-18). Feeling “makes your life and your relationships and deeds feel
unique and gives them a definite value” (pp. IV-18–IV-19). Your beloved is like no
one else. Your own death is a face behind a door. And the meaning of ‘civilian deaths
this month due to the war were 20,964’ is unfathomable horror—not a number.

In short, there have been many who have questioned an exclusively statistical
view of our subject matter, and their voices should be considered too as you proceed
with your study of what has become the predominant, but not exclusive, means of
doing psychology research.

B O X 2 – 2 Gender, Ethnicity, and Math Performance
From time to time, someone tries to argue that because
some groups of people score better on math tests and
make careers out of mathematics, these groups have a
genetic advantage in math (or statistics). Other groups
are said or implied to be innately inferior at math. The
issue comes up about gender, about racial and ethnic
groups, and of course in arguments about overall intelli-
gence as well as math. There’s little evidence for such
genetic differences (a must-see article is Block, 1995),
but the stereotypes persist.

The impact of these stereotypes has been well estab-
lished in research by Steele and his colleagues (1997),
who have done numerous studies on what they call “stereo-
type threat.” This phenomenon occurs when a negative
stereotype about a group you belong to becomes relevant
to you because of the situation you are in, like taking a
math test, and provides an explanation for how you will
behave. A typical experiment creating stereotype threat
(Spencer et al., 1999) involved women taking a difficult
math test. Half were told that men generally do better on
the test, and the other half that women generally do equally
well. Those who were told that women do worse did
indeed score substantially lower than the other group. In
the other condition, there was no difference. (In fact, in

two separate studies, men performed a little worse when
they were told there was no gender difference, as if they
had lost some of their confidence.)

The same results occur when African Americans are
given parts of the graduate record examination. They do
fine on the test when they are told no racial differences in
the scores have been found, and they do worse when they
are told that such differences have been found (Steele,
1997).

Stereotype threat has also been found to occur in the
United States for Latinos (Gonzales et al., 2002) and the
poor (Croizet & Claire, 1998). Many lines of research in-
dicate that prejudices, not genetics, are the probable
cause of differences in test scores between groups. Al-
though some researchers (Rushton & Jensen, 2005) con-
tinue to argue for genetic differences, the evidence is still
substantial that stereotype threat plays the main role in
lower test scores (Suzuki & Aronson, 2005). For exam-
ple, the same difference of 15 IQ points between a domi-
nant and minority group has been found all over the
world, even when there is no genetic difference between
the groups, and in cases where opportunities for a group
have changed, such as when they emigrate, differences
have rapidly disappeared (Block, 1995).

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54 Chapter 2

If groups such as women and African Americans are
not inherently inferior but perform worse on tests, what
might be the reasons? The usual explanation is that they
have internalized the “superior” group’s prejudices. Steele
thinks the problem might not be so internal but may have
to do with the situation. The stigmatized groups perform
worse when they know that’s what is expected—when they
experience the threat of being stereotyped. They either
become too anxious or give up and avoid the subject.

What Can You Do for Yourself?
So, do you feel you belong to a group that is expected to
do poorly at math? (Perhaps the group of “math dumb-
bells” in the class?) What can you do to get out from under
the shadow of stereotype threat as you take this course?

First, care about learning statistics. Don’t discount it
to save your self-esteem. Fight for your right to know
this subject. Consider these words from the former presi-
dent of the Mathematics Association of America:

The paradox of our times is that as mathematics becomes
increasingly powerful, only the powerful seem to benefit
from it. The ability to think mathematically—broadly
interpreted—is absolutely crucial to advancement in vir-
tually every career. Confidence in dealing with data,
skepticism in analyzing arguments, persistence in pene-
trating complex problems, and literacy in communicating
about technical matters are the enabling arts offered by
the new mathematical sciences. (Steen, 1987, p. xviii)

Second, once you care about succeeding at statistics,
realize you are going to be affected by stereotype threat.
Think of it as a stereotype-induced form of test anxiety
and work on it that way (see Box 1–2).

Third, root out the effects of that stereotype in your-
self as much as you can. It takes some effort. That’s why
we are spending time on it here. Research on stereotypes
shows that they can be activated without our awareness
(Fiske, 1998), even when we are otherwise low in preju-
dice or a member of the stereotyped group.

Some Points to Think About
For women, yes, the very top performers tend to be male,
but the differences are slight, and the lowest performers
are not more likely to be female. Indeed, gender differ-
ences on test performance have been declining (National
Center for Education Statistics, 2001). Tobias (1982) cites
numerous studies for why women might not make it to
the very top in math. For example, in a study of students
identified by a math talent search, it was found that few

parents arranged for their daughters to be coached before
the talent exams. Sons were almost invariably coached. In
another study, parents of mathematically gifted girls were
not even aware of their daughters’ abilities, whereas par-
ents of boys invariably were. In general, girls tend to
avoid higher math classes, according to Tobias, because
parents, peers, and even teachers often advise them
against pursuing too much math. Indeed, mothers’ views
of their child’s math abilities are strong predictors of their
later performance (Bleeker & Jacobs, 2004). Girls fre-
quently outperform boys in math, yet still greatly under-
estimate their abilities (Heller & Ziegler, 1996). So, even
though women are earning more PhDs in math than ever
before, it is not surprising that math is the field with the
highest dropout rate for women.

We checked the grades in our own introductory statis-
tics classes and found no reliable difference for gender.
More generally, Schram (1996) analyzed results of 13 in-
dependent studies of performance in college statistics and
found an overall average difference of almost exactly zero
(the slight direction of difference favored females). Steele
(1997) also found that the grades of African Americans,
for example, rose substantially when they were enrolled
in a transition-to-college program emphasizing that they
were the cream of the crop and much was expected of
them. Meanwhile, African American students at the same
school who were enrolled in a remedial program for mi-
norities received considerable attention, but their grades
improved very little and many more of them dropped out
of school than in the other group. Steele argues that the
very idea of a remedial program exposed those students to
a subtle stereotype threat.

Cognitive research on stereotype threat has demon-
strated that it most affects math problems relying on long-
term memory and spills over into subsequent tasks not
normally affected by stereotype threat (Beilock et al., 2007).

Another point to ponder is a study cited by Tobias
(1995) comparing students in Asia and the United
States on an international mathematics test. The U.S. stu-
dents were thoroughly outperformed, but more important
was why: Interviews revealed that Asian students saw
math as an ability fairly equally distributed among people
and thought that differences in performance were due to
hard work. Contrarily, U.S. students thought some people
are just born better at math; so hard work matters little.

In short, our culture’s belief that “math just comes
naturally to some people” could be holding you back.
But then, doing well in this course may even be more sat-
isfying for you than for others.

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Central Tendency and Variability 55

Central Tendency and Variability
in Research Articles
The mean and the standard deviation are very commonly reported in research arti-
cles. However, the mode, median, and variance are only occasionally reported.
Sometimes the mean and standard deviation are included in the text of an article. For
our dreams example, the researcher might write, “The mean number of dreams in the
last week for the 10 students was 6.00 ( ).” Means and standard deviations
are also often listed in tables, especially if a study includes several groups or several
different variables. For example, Selwyn (2007) conducted a study of gender-related
perceptions of information and communication technologies (such as games ma-
chines, DVD players, and cell phones). The researcher asked 406 college students in
Wales to rate 8 technologies in terms of their level of masculinity or femininity. The
students rated each technology using a 7-point response scale, from for very fem-
inine to for very masculine, with a midpoint of 0 for neither masculine or femi-
nine. Table 2–5 (reproduced from Selwyn’s article) shows the mean, standard
deviation, and variance of the students’ ratings of each technology. As the table
shows, games machines were rated as being more masculine than feminine, and land-
line telephones were rated as being slightly more feminine than masculine. Notice
that Table 2–5 is one of those rare examples where the variance is shown (usually just
the standard deviation is given). Overall, the table provides a useful summary of the
descriptive results of the study. In another part of the study, Selwyn compared
women’s and men’s perceptions of the masculinity or femininity of different aspects
of computers and computing. We will describe those results in Chapter 8; so be sure
to look out for them!

Another interesting example is shown in Table 2–6 (reproduced from Norcross
et al., 2005). The table shows the application and enrollment statistics for psychology
doctoral programs in the United States, broken down by area of psychology and by
year (1973, 1979, 1992, and 2003). The table does not give standard deviations, but
it does give both means and medians. For example, in 2003 the mean number of ap-
plicants to doctoral counseling psychology programs was 71.0, but the median was
only 59. This suggests that some programs had very high numbers of applicants that

+ 3
– 3

SD = 2.57

Table 2–5 Mean Scores for Each Technology

N Mean S.D. Variance

Games machine (e.g., Playstation) 403 1.92 1.00 .98

DVD Player 406 .44 .85 .73

Personal Computer (PC) 400 .36 .82 .68

Digital radio (DAB) 399 .34 .99 .98

Television set 406 .26 .78 .62

Radio 404 .81 .65

Mobile phone 399 .88 .77

Landline telephone 404 1.03 1.07

Note: Mean scores range from �3 (very feminine) to +3 (very masculine). The midpoint score of .0 denotes “neither masculine
nor feminine.”
Source: Selwyn, N. (2007). Hi-tech = guy-tech? An exploration of undergraduate students’ gendered perceptions of information
and communication technologies. Sex Roles, 56, 525–536. Copyright © 2007. Reprinted by permission of Springer Science and
Business Media.

– .77
– .19
– .01

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Central Tendency and Variability 57

skewed the distribution. In fact, you can see from the table that for almost every kind
of program and for both applications and enrollments, the means are typically higher
than the medians. You may also be struck by just how competitive it is to get into
doctoral programs in many areas of psychology. It is our experience that one of the
factors that makes a lot of difference is doing well in statistics courses!

1. The mean is the most commonly used measure of central tendency of a distrib-
ution of scores. The mean is the ordinary average—the sum of the scores divided
by the number of scores. In symbols, .

2. Other, less commonly used ways of describing the central tendency of a distribu-
tion of scores are the mode (the most common single value) and the median (the
value of the middle score when all the scores are lined up from lowest to highest).

3. The variability of a group of scores can be described by the variance and the
standard deviation.

4. The variance is the average of the squared deviation of each score from the
mean. In symbols, . The sum of squared deviations,

, is also symbolized as SS. Thus .
5. The standard deviation is the square root of the variance. In symbols,

It is approximately the average amount that scores differ from
the mean.

6. There have always been a few psychologists who have warned against statistical
methodology because in the process of creating averages, knowledge about the
individual case is lost.

7. Means and standard deviations are often given in research articles in the text or
in tables.

SD = 2SD2.
SD2 = SS>N©(X – M)2

SD2 = [©(X – M)2]> N

M = (©X)>N

Summary

Figuring the Mean
Find the mean for the following scores: 8, 6, 6, 9, 6, 5, 6, 2.

Answer
You can figure the mean using the formula or the steps.

Using the formula: .
Using the steps:

❶ Add up all the scores. .
❷ Divide this sum by the number of scores. .48> 8 = 6

8 + 6 + 6 + 9 + 6 + 5 + 6 + 2 = 48

M = (©X)> N = 48> 8 = 6

Example Worked-Out Problems

Key Terms

central tendency (p. 34)
mean (M) (pp. 34, 35)
Σ (sum of ) (p. 35)
X (p. 35)
N (number of scores) (p. 36)
mode (p. 37)
median (p. 39)

outlier (p. 39)
variance (SD2 ) (p. 44)
deviation score (p. 44)
squared deviation score (p. 44)
sum of squared deviations (sum of

squares) (SS ) (pp. 44, 46)
standard deviation (SD) (p. 45)

SD2 (p. 46)
SD (p. 46)
computational formula (p. 49)
definitional formula (p. 49)

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58 Chapter 2

Finding the Median
Find the median for the following scores: 1, 7, 4, 2, 3, 6, 2, 9, 7.

Answer
❶ Line up all the scores from lowest to highest. 1, 2, 2, 3, 4, 6, 7, 7, 9.
❷ Figure how many scores there are to the middle score by adding 1 to the

number of scores and dividing by 2. There are 9 scores; so the middle score is
the result of adding 1 to 9 and then dividing by 2, which is 5. The middle score
is the fifth score.

❸ Count up to the middle score or scores. The fifth score from the bottom is 4;
so the median is 4.

Figuring the Sum of Squares and the Variance
Find the sum of squares and the variance for the following scores: 8, 6, 6, 9, 6, 5, 6,
2. (These are the same scores used in the previous example for the mean: .)

Answer
You can figure the sum of squares and the variance using the formulas or the steps.

Using the formulas:

Table 2–7 shows the figuring, using the following steps:

❶ Subtract the mean from each score.
❷ Square each of these deviation scores.
❸ Add up the squared deviation scores. This gives the sum of squares (SS).
❹ Divide the sum of squared deviations by the number of scores. This gives the

variance (SD2).

SD2 = SS>N = 30> 8 = 3.75.
= 30
= 4 + 0 + 0 + 9 + 0 + 1 + 0 + 16

= 22 + 02 + 02 + 32 + 02 + – 12 + 02 + – 42
+ (9 – 6)2 + (6 – 6)2 + (5 – 6)2 + (6 – 6)2 + (2 – 6)2

SS = ©(X – M)2 = (8 – 6)2 + (6 – 6)2 + (6 – 6)2

M = 6

Table 2–7 Figuring for Example Worked-Out Problem
for the Sum of Squares and Variance Using Steps

❶ ❷

Score Mean Deviation Squared Deviation

8 6 2 4
6 6 0 0
6 6 0 0
9 6 3 9
6 6 0 0

5 6 1

6 6 0 0

2 6 16

❸
❹ Variance = 30/ 8 = 3.75

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Central Tendency and Variability 59

Figuring the Standard Deviation
Find the standard deviation for the following scores: 8, 6, 6, 9, 6, 5, 6, 2. (These are the
same scores used above for the mean, sum of squares, and variance. .)

Answer
You can figure the standard deviation using the formula or the steps.

Using the formula: .
Using the steps:

❶ Figure the variance. The variance (from above) is 3.75.
❷ Take the square root. The square root of 3.75 is 1.94.

Outline for Writing Essays on Finding the Mean, Variance,
and Standard Deviation

1. Explain that the mean is a measure of the central tendency of a group of scores.
Mention that the mean is the ordinary average, that is, the sum of the scores
divided by the number of scores.

2. Explain that the variance and standard deviation both measure the amount of
variability (or spread) among a group of scores.

3. The variance is the average of each score’s squared difference from the mean.
Describe the steps for figuring the variance.

4. Roughly speaking, the standard deviation is the average amount that scores dif-
fer from the mean. Explain that the standard deviation is directly related to the
variance and is figured by taking the square root of the variance.

SD = 2SD2 = 23.75 = 1.94

SD2 = 3.75

These problems involve figuring. Most real-life statistics problems are done on a
computer with special statistical software. Even if you have such software, do these
problems by hand to ingrain the method in your mind. To learn how to use a comput-
er to solve statistics problems like those in this chapter, refer to the Using SPSS sec-
tion at the end of this chapter and the Study Guide and Computer Workbook that
accompanies this text.

All data are fictional unless an actual citation is given.

Set I (for Answers to Set I Problems, see pp. 674–675)
1. For the following scores, find the (a) mean, (b) median, (c) sum of squared

deviations, (d) variance, and (e) standard deviation:

32, 28, 24, 28, 28, 31, 35, 29, 26

2. For the following scores, find the (a) mean, (b) median, (c) sum of squared
deviations, (d) variance, and (e) standard deviation:

6, 1, 4, 2, 3, 4, 6, 6

3. For the following scores, find the (a) mean, (b) median, (c) sum of squared
deviations, (d) variance, and (e) standard deviation:

2.13, 6.01, 3.33, 5.78

Practice Problems

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60 Chapter 2

4. Here are the noon temperatures (in degrees Celsius) in a particular Canadian
city on Boxing Day (usually December 26) for the 10 years from 1998 through
2007: , , , 0, , , , , and . Describe the typical tem-
perature and the amount of variation to a person who has never had a course in
statistics. Give three ways of describing the representative temperature and two
ways of describing its variation, explaining the differences and how you figured
each. (You will learn more if you try to write your own answer first, before read-
ing our answer at the back of the book.)

5. A researcher is studying the amygdala (a part of the brain involved in emotion). Six
participants in a particular fMRI (brain scan) study are measured for the increase in
activation of their amygdala while they are viewing pictures of violent scenes. The
activation increases are .43, .32, .64, .21, .29, and .51. Figure the (a) mean and (b)
standard deviation for these six activation increases. (c) Explain what you have
done and what the results mean to a person who has never had a course in statistics.

6. Describe and explain the location of the mean, mode, and median for a normal
curve.

7. A researcher studied the number of anxiety attacks recounted over a two-week
period by 30 people in psychotherapy for an anxiety disorder. In an article de-
scribing the results of the study, the researcher reports: “The mean number of
anxiety attacks was 6.84 ( ).” Explain these results to a person who
has never had a course in statistics.

8. In a study by Gonzaga et al. (2001), romantic couples answered questions about
how much they loved their partner and also were videotaped while revealing
something about themselves to their partner. The videotapes were later rated by
trained judges for various signs of affiliation. Table 2–8 (reproduced from their
Table 2) shows some of the results. Explain to a person who has never had a
course in statistics the results for self-reported love for the partner and for the
number of seconds “leaning toward the partner.”

Set II
9. (a) Describe and explain the difference between the mean, median, and mode. (b)

Make up an example (not in the book or in your lectures) in which the median
would be the preferred measure of central tendency.

SD = 3.18

– 24- 13- 9- 5- 8- 1,- 1- 4- 5

Table 2–8 Mean Levels of Emotions and Cue Display in Study 1

Women ( ) Men ( )

Indicator M SD M SD

Emotion reports

Self-reported love 5.02 2.16 5.11 2.08

Partner-estimated love 4.85 2.13 4.58 2.20

Affiliation-cue display

Affirmative head nods 1.28 2.89 1.21 1.91

Duchenne smiles 4.45 5.24 5.78 5.59

Leaning toward partner 32.27 20.36 31.36 21.08

Gesticulation 0.13 0.40 0.25 0.77

Note: Emotions are rated on a scale of 0 (none) to 8 (extreme). Cue displays are shown as mean
seconds displayed per 60 s.
Source: Gonzaga, G. C., Keltner, D., Londahl, E. A., & Smith, M. D. (2001). Love and the commitment
problem in romantic relationships and friendship. Journal of Personality and Social Psychology, 81,
247–262. Published by the American Psychological Association. Reprinted with permission.

n � 60n � 60

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Central Tendency and Variability 61

10. (a) Describe the variance and standard deviation. (b) Explain why the standard
deviation is more often used as a descriptive statistic than the variance.

11. For the following scores, find the (a) mean, (b) median, (c) sum of squared
deviations, (d) variance, and (e) standard deviation:

2, 2, 0, 5, 1, 4, 1, 3, 0, 0, 1, 4, 4, 0, 1, 4, 3, 4, 2, 1, 0

12. For the following scores, find the (a) mean, (b) median, (c) sum of squared
deviations, (d) variance, and (e) standard deviation:

1,112; 1,245; 1,361; 1,372; 1,472

13. For the following scores, find the (a) mean, (b) median, (c) sum of squared
deviations, (d) variance, and (e) standard deviation:

3.0, 3.4, 2.6, 3.3, 3.5, 3.2

14. For the following scores, find the (a) mean, (b) median, (c) sum of squared
deviations, (d) variance, and (e) standard deviation:

8, –5, 7, –10, 5

15. Make up three sets of scores: (a) one with the mean greater than the median,
(b) one with the median and the mean the same, and (c) one with the mode greater
than the median. (Each made-up set of scores should include at least five scores.)

16. A psychologist interested in political behavior measured the square footage of the
desks in the official office of four U.S. governors and of four chief executive offi-
cers (CEOs) of major U.S. corporations. The figures for the governors were 44, 36,
52, and 40 square feet. The figures for the CEOs were 32, 60, 48, and 36 square
feet. (a) Figure the means and standard deviations for the governors and for the
CEOs. (b) Explain, to a person who has never had a course in statistics, what you
have done. (c) Note the ways in which the means and standard deviations differ,
and speculate on the possible meaning of these differences, presuming that they
are representative of U.S. governors and large corporations’ CEOs in general.

17. A developmental psychologist studies the number of words that seven infants
have learned at a particular age. The numbers are 10, 12, 8, 0, 3, 40, and 18. Fig-
ure the (a) mean, (b) median, and (c) standard deviation for the number of words
learned by these seven infants. (d) Explain what you have done and what the re-
sults mean to a person who has never had a course in statistics.

18. Describe and explain the location of the mean, mode, and median of a distribu-
tion of scores that is strongly skewed to the left.

19. You figure the variance of a distribution of scores to be – 4.26. Explain why your
answer cannot be correct.

20. A study involves measuring the number of days absent from work for 216 em-
ployees of a large company during the preceding year. As part of the results, the
researcher reports, “The number of days absent during the preceding year
( ) was . . . .” Explain what is written in parentheses to a
person who has never had a course in statistics.

21. Payne (2001) gave participants a computerized task in which they first see a
face and then a picture of either a gun or a tool. The task was to press one button
if it was a tool and a different one if it was a gun. Unknown to the participants
while they were doing the study, the faces served as a “prime” (something that
starts you thinking a particular way); half the time they were of a black person
and half the time of a white person. Table 2–9 shows the means and standard de-
viations for reaction times (the time to decide if the picture is of a gun or a tool)
after either a black or white prime. (In Experiment 2, participants were told to

M = 9.21; SD = 7.34

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62 Chapter 2

decide as fast as possible.) Explain the results to a person who has never had a
course in statistics. (Be sure to explain some specific numbers as well as the
general principle of the mean and standard deviation.)

Table 2–9 Mean Reaction Times (in Milliseconds) in Identifying Guns
and Tools in Experiments 1 and 2

Prime

Black White

Target M SD M SD

Experiment 1

Gun 423 64 441 73

Tool 454 57 446 60

Experiment 2

Gun 299 28 295 31

Tool 307 29 304 29

Source: Payne, B. K. (2001). Prejudice and perception: The role of automatic and controlled processes
in misperceiving a weapon. Journal of Personality and Social Psychology, 81, 181–192. Published
by the American Psychological Association. Reprinted with permission.

Using SPSS

The U in the following steps indicates a mouse click. (We used SPSS version 15.0
to carry out these analyses. The steps and output may be slightly different for other
versions of SPSS.)

Finding the Mean, Mode, and Median
❶ Enter the scores from your distribution in one column of the data window.
❷ U Analyze.
❸ U Descriptive statistics.
❹ U Frequencies.
➎ U on the variable for which you want to find the mean, mode, and median, and

then U the arrow.
➏ Statistics.
❼ U Mean, U Median, U Mode, U Continue.
❽ Optional: To instruct SPSS not to produce a frequency table, U the box labeled

Display frequency tables (this unchecks the box).
❾ U OK.

Practice the steps above by finding the mean, mode, and median for the number
of dreams example at the start of the chapter (the scores are 7, 8, 8, 7, 3, 1, 6, 9, 3, 8).
Your output window should look like Figure 2–14. (If you instructed SPSS not to
show the frequency table, your output will show only the mean, median, and mode.)

Finding the Variance and Standard Deviation
As mentioned earlier in the chapter, most calculators and computer software—
including SPSS—calculate the variance and standard deviation using a formula that
involves dividing by N – 1 instead of N. So, if you request the variance and standard

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Central Tendency and Variability 63

deviation directly from SPSS (for example, by clicking Variance and Std. deviation
in Step ❼), the answers provided by SPSS will be different from the answers in this
chapter.5 The following steps show you how to use SPSS to figure the variance and
standard deviation using the dividing-by-N method you learned in this chapter. It is
easier to learn these steps using actual numbers; so we will use the number of dreams
example again.

❶ Enter the scores from your distribution in one column of the data window (the
scores are 7, 8, 8, 7, 3, 1, 6, 9, 3, 8). We will call this variable “dreams.”

❷ Find the mean of the scores by following the preceding steps for Finding the
Mean, Mode, and Median. The mean of the dreams variable is 6.

❸ You are now going to create a new variable that shows each score’s squared
deviation from the mean. U Transform, U Compute Variable. You could call
the new variable any name you want, but we will call it “sqdev” (for “squared

Figure 2–14 Using SPSS to find the mean, median, and mode for the number of dreams
example.

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64 Chapter 2

deviation”). So, write sqdev in the box labeled Target Variable. You are now
going to tell SPSS how to figure sqdev. In the box labeled Numeric Expression,
write (dreams ) * (dreams ). (The asterisk is how you show multiplication
in SPSS.) As you can see, this formula takes each score’s deviation score and
multiplies it by itself to give the squared deviation score. Your Compute Variable
window should look like Figure 2–15. U OK. You will see that a new variable
called sqdev has been added to the data window (see Figure 2–16). The scores
are the squared deviations of each score from the mean.

❹ As you learned in this chapter, the variance is figured by dividing the sum of the
squared deviations by the number of scores. This is the same as taking the mean
of the squared deviation scores. So, to find the variance of the dreams scores,
follow the steps shown earlier to find the mean of the sqdev variable. This
comes out to 6.60; so the variance of the dreams scores is 6.60.

➎ To find the standard deviation, use a calculator to find the square root of 6.60,
which is 2.57.

If you were conducting an actual research study, you would most likely request
the variance and standard deviation directly from SPSS. However, for our purpose in
this chapter (describing the variation in a group of scores), the steps we just outlined
are entirely appropriate.

– 6- 6

Figure 2–15 SPSS compute variable window for Step ❸ of finding the variance and
standard deviation for the number of dreams example.

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Central Tendency and Variability 65

Figure 2–16 SPSS data window after Step ❸ for finding the variance and standard
deviation for the number of dreams example.

1. In more formal, mathematical statistics writing, the symbols can be more com-
plex. This complexity allows formulas to handle intricate situations without
confusion. However, in books on statistics for psychologists, even fairly ad-
vanced texts, the symbols are kept simple. The simplified form rarely creates
ambiguities in the kinds of statistical formulas psychologists use.

2. This section focuses on the variance and standard deviation as indicators of
spread, or variability. Another way to describe the spread of a group of scores is
in terms of the range—the highest score minus the lowest score. Suppose that in
a particular class the oldest student is 39 years of age and the youngest is 19; the
range is 20 (that is, ). Psychology researchers rarely use the range
because it is such an imprecise way to describe the spread; it does not take into
account how clumped together the scores are within the range.

3. Why don’t statisticians use the deviation scores themselves, make all deviations
positive, and just use their average? In fact, the average of the deviation scores
(treating all deviations as positive) has a formal name—the average deviation or
mean deviation. This procedure was actually used in the past, and some psy-
chologists have noted subtle advantages of the average deviation (Catanzaro &
Taylor, 1996). However, the average deviation does not work out very well as
part of more complicated statistical procedures.

39 – 19 = 20

Chapter Notes

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66 Chapter 2

4. It is important to remember that the standard deviation is not exactly the average
amount that scores differ from the mean. To be precise, the standard deviation is
the square root of the average of scores’ squared deviations from the mean. This
squaring, averaging, and then taking the square root gives a slightly different re-
sult from simply averaging the scores’ deviations from the mean. Still, the result
of this approach has technical advantages that outweigh the slight disadvantage
of giving only an approximate description of the average variation from the
mean (see Chapter Note 3).

5. Note that if you request the variance from SPSS, you can convert it to the vari-
ance as we figure it in this chapter by multiplying the variance from SPSS by

(that is, the number of scores minus 1) and then dividing the result by N
(the number of scores). (That is the variance as we are figuring it in this chapter �
SPSS’s variance multiplied by [ ]�N.) Taking the square root of the resulting
value will give you the standard deviation (using the formula you learned in this
chapter). We use a slightly longer approach to figuring the variance and standard
deviation in order to show you how to create new variables in SPSS.

N – 1
N – 1
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Name:

Chapter 3 Instructions

Practice Problems 14, 15, 22, & 25

Due Week 3 Day 6 (Sunday)

Follow the instructions below to submit your answers for Chapter 3 Practice Problem 14, 15, 22 & 25.

1. Save Chapter 3 Instructions to your computer.

2. Type your answers into the shaded boxes below. The boxes will expand as you type your answers.

3. Resave this form to your computer with your answers filled-in.

Below is an explanation of the symbols for Chapter 3 Practice Problem 14 and 15.

Z = Z Score

R = Raw score

Read each question in your text book and then type your answers for Chapter 3, Practice Problem 14 & 15 in the shaded boxes below. Please record only your answers. It is not necessary to show your work. Round your answers to 2 decimal places.

14a. Z =

14b. Z =

14c. Z =

14d. R =

14e. R =

14f. R =

14g. R =

15. Verbal Ability Z =

Quantitative Ability Z =

ability is stronger.

22a.

22b.

For practice problem 25, please represent your answers as decimals. For example: .25 or .01 etc.

25a.

25b.

25c.

25d.

25e.

CHAPTER 3

6767

Some Key Ingredients
for Inferential Statistics

Z Scores, the Normal Curve, Sample
versus Population, and Probability

Chapter Outline

✪ Z Scores 68

✪ The Normal Curve 73

✪ Sample and Population 83

✪ Probability 88

✪ Controversies: Is the Normal Curve
Really So Normal? and Using
Nonrandom Samples 93

✪ Z Scores, Normal Curves, Samples
and Populations, and Probabilities
in Research Articles

Ordinarily, psychologists conduct research to test a theoretical principle or theeffectiveness of a practical procedure. For example, a psychophysiologist mightmeasure changes in heart rate from before to after solving a difficult problem.
The measurements are then used to test a theory predicting that heart rate should change
following successful problem solving. An applied social psychologist might examine

✪ Advanced Topic: Probability Rules
and Conditional Probabilities 96

✪ Summary 97

✪ Key Terms 98

✪ Example Worked-Out Problems 99

✪ Practice Problems 102

✪ Using SPSS 105

✪ Chapter Notes 106

T I P F O R S U C C E S S
Before beginning this chapter, be
sure you have mastered the mater-
ial in Chapter 1 on the shapes of
distributions and the material in
Chapter 2 on the mean and stan-
dard deviation.

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68 Chapter 3

the effectiveness of a program of neighborhood meetings intended to promote water
conservation. Such studies are carried out with a particular group of research partici-
pants. But researchers use inferential statistics to make more general conclusions about
the theoretical principle or procedure being studied. These conclusions go beyond the
particular group of research participants studied.

This chapter and Chapters 4, 5, and 6 introduce inferential statistics. In this
chapter, we consider four topics: Z scores, the normal curve, sample versus popula-
tion, and probability. This chapter prepares the way for the next ones, which are
more demanding conceptually.

Z Scores
In Chapter 2, you learned how to describe a group of scores in terms and the mean
and variation around the mean. In this section you learn how to describe a particular
score in terms of where it fits into the overall group of scores. That is, you learn how
to use the mean and standard deviation to create a Z score; a Z score describes a score
in terms of how much it is above or below the average.

Suppose you are told that a student, Jerome, is asked the question, “To what extent
are you a morning person?” Jerome responds with a 5 on a 7-point scale, where
not at all and extremely. Now suppose that we do not know anything about how
other students answer this question. In this situation, it is hard to tell whether Jerome is
more or less of a morning person in relation to other students. However, suppose that
we know for students in general, the mean rating (M) is 3.40 and the standard deviation
(SD) is 1.47. (These values are the actual mean and standard deviation that we found
for this question in a large sample of statistics students from eight different universities
across the United States and Canada.) With this knowledge, we can see that Jerome is
more of a morning person than is typical among students. We can also see that Jerome
is above the average (1.60 units more than average; that is, ) by a bit
more than students typically vary from the average (that is, students typically vary by
about 1.47, the standard deviation). This is all shown in Figure 3–1.

What Is a Z Score?
A Z score makes use of the mean and standard deviation to describe a particular
score. Specifically, a Z score is the number of standard deviations the actual score is
above or below the mean. If the actual score is above the mean, the Z score is posi-
tive. If the actual score is below the mean, the Z score is negative. The standard
deviation now becomes a kind of yardstick, a unit of measure in its own right.

In our example, Jerome has a score of 5, which is 1.60 units above the mean of 3.40.
One standard deviation is 1.47 units; so Jerome’s score is a little more than 1 standard

5 – 3.40 = 1.6

7 =
1 =

.46

Mean

1.93

3.40

4.87

6.3

Jerome’s
score
(5)

Figure 3–1 Score of one student, Jerome, in relation to the overall distribution on the
measure of the extent to which students are morning people.

Z score number of standard deviations
that a score is above (or below, if it is
negative) the mean of its distribution; it
is thus an ordinary score transformed so
that it better describes the score’s location
in a distribution.

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Some Key Ingredients for Inferential Statistics 69

deviation above the mean. To be precise, Jerome’s Z score is (that is, his score of
5 is 1.09 standard deviations above the mean).Another student, Michelle, has a score of
2. Her score is 1.40 units below the mean. Therefore, her score is a little less than 1 stan-
dard deviation below the mean (a Z score of ). So, Michelle’s score is below the
average by about as much as students typically vary from the average.

Z scores have many practical uses.As you will see later in this chapter, they are es-
pecially useful for showing exactly where a particular score falls on the normal curve.

Z Scores as a Scale
Figure 3–2 shows a scale of Z scores lined up against a scale of raw scores for our
example of the degree to which students are morning people. A raw score is an ordi-
nary score as opposed to a Z score. The two scales are something like a ruler with
inches lined up on one side and centimeters on the other.

Changing a number to a Z score is a bit like converting words for measurement
in various obscure languages into one language that everyone can understand—inches,
cubits, and zingles (we made up that last one), for example, into centimeters. It is a
very valuable tool.

Suppose that a developmental psychologist observed 3-year-old David in a lab-
oratory situation playing with other children of the same age. During the observa-
tion, the psychologist counted the number of times David spoke to the other children.
The result, over several observations, is that David spoke to other children about
8 times per hour of play. Without any standard of comparison, it would be hard to
draw any conclusions from this. Let’s assume, however, that it was known from pre-
vious research that under similar conditions, the mean number of times children
speak is 12, with a standard deviation of 4. With that information, we can see that
David spoke less often than other children in general, but not extremely less often.
David would have a Z score of ( and , thus a score of 8 is 1 SD
below M), as shown in Figure 3–3.

Suppose Ryan was observed speaking to other children 20 times in an hour. Ryan
would clearly be unusually talkative, with a Z score of (see Figure 3–3). Ryan
speaks not merely more than the average but more by twice as much as children tend
to vary from the average!

SD = 4M = 12

-1

– .95

+1.09

Z score:

Raw score: .46

−2

1.93

−1

3.40
0
4.87

6.34

Figure 3–2 Scales of Z scores and raw scores for the example of the extent to which
students are morning people.

Z score:

Times spoken per hour: 0

−3

4
−1

+3−2

David Ryan

Figure 3–3 Number of times each hour that two children spoke, shown as raw scores
and Z scores.

raw score ordinary score (or any num-
ber in a distribution before it has been
made into a Z score or otherwise trans-
formed).

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70 Chapter 3

Formula to Change a Raw Score to a Z Score
A Z score is the number of standard deviations by which the raw score is above or
below the mean. To figure a Z score, subtract the mean from the raw score, giving
the deviation score. Then divide the deviation score by the standard deviation. The
formula is

(3–1)

For example, using the formula for David, the child who spoke to other children
8 times in an hour (where the mean number of times children speak is 12 and the
standard deviation is 4),

Steps to Change a Raw Score to a Z Score

❶ Figure the deviation score: subtract the mean from the raw score.

❷ Figure the Z score: divide the deviation score by the standard deviation.

Using these steps for David, the child who spoke with other children 8 times in
an hour,

❶ Figure the deviation score: subtract the mean from the raw score.
❷ Figure the Z score: divide the deviation score by the standard deviation.

Formula to Change a Z Score to a Raw Score
To change a Z score to a raw score, the process is reversed: multiply the Z score by
the standard deviation and then add the mean. The formula is

(3–2)

Suppose a child has a Z score of 1.5 on the number of times spoken with another
child during an hour. This child is 1.5 standard deviations above the mean. Because
the standard deviation in this example is 4 raw score units (times spoken), the child
is 6 raw score units above the mean, which is 12. Thus, 6 units above the mean is 18.
Using the formula,

Steps to Change a Z Score to a Raw Score

❶ Figure the deviation score: multiply the Z score by the standard deviation.

❷ Figure the raw score: add the mean to the deviation score.

Using these steps for the child with a Z score of 1.5 on the number of times
spoken with another child during an hour:

❶ Figure the deviation score: multiply the Z score by the standard deviation.

❷ Figure the raw score: add the mean to the deviation score. 6 + 12 = 18.
1.5 * 4 = 6.

X = (Z) (SD) + M = (1.5) (4) + 12 = 6 + 12 = 18

X = (Z) (SD) + M

-4/4 = -1.

8 – 12 = -4.

Z =
8 – 12

4
=

-4
4

= -1

Z =
X – M

A Z score is the raw score
minus the mean, divided by
the standard deviation.

The raw score is the Z score
multiplied by the standard
deviation, plus the mean.

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Some Key Ingredients for Inferential Statistics 71

Additional Examples of Changing Z Scores
to Raw Scores and Vice Versa
Consider again the example from the start of the chapter in which students were
asked the extent to which they were a morning person. Using a scale from 1 (not at
all ) to 7 (extremely), the mean was 3.40 and the standard deviation was 1.47. Sup-
pose a student’s raw score is 6. That student is well above the mean. Specifically,
using the formula,

That is, the student’s raw score is 1.77 standard deviations above the mean
(see Figure 3–4, Student 1). Using the 7-point scale (from to

), to what extent are you a morning person? Now figure the Z score for
your raw score.

Another student has a Z score of , a score well below the mean. (This stu-
dent is much less of a morning person than is typically the case for students.) You
can find the exact raw score for this student using the formula

That is, the student’s raw score is 1.00 (see Figure 3–4, Student 2).
Let’s also consider some examples from the study of students’ stress ratings.

The mean stress rating of the 30 statistics students (using a 0–10 scale) was 6.43 (see
Figure 2–3), and the standard deviation was 2.56. Figure 3–5 shows the raw score
and Z score scales. Suppose a student’s stress raw score is 10. That student is well
above the mean. Specifically, using the formula

Z =
X – M

SD
=

10 –

6.43

2.56

=
3.57

2.56
= 1.39

X = (Z) (SD) + M = (-1.63) (1.47) + 3.40 = -2.40 + 3.40 = 1.00

-1.63

extremely
7 =1 = not at all

Z =
X – M
SD
=

6 – 3.40
1.47

=
2.60

1.47
= 1.77

Z score:
Raw score: .46
−2
1.93
−1
3.40
0
4.87
+1
6.34
+2

(1.00)
Student 2

(6.00)
Student 1

Figure 3–4 Scales of Z scores and raw scores for the example of the extent to which
students are morning people, showing the scores of two sample students.

Z score:

(2.00)
Student 2

(10.00)
Student 1

Stress rating:

−3

1.31

−1
6.43
0

8.99

14.11

+3−2

3.87 11.55

−1.25

Figure 3–5 Scales of Z scores and raw scores for 30 statistics students’ ratings of their
stress level, showing the scores of two sample students. (Data based on Aron et al., 1995.)

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72 Chapter 3

The student’s stress level is 1.39 standard deviations above the mean (see Figure
3–5, Student 1). On a scale of 0–10, how stressed have you been in the last 21⁄2
weeks? Figure the Z score for your raw stress score.

Another student has a Z score of , a stress level well below the mean. You
can find the exact raw stress score for this student using the formula

That is, the student’s raw stress score is 2.00 (see Figure 3–5, Student 2).

The Mean and Standard Deviation of Z Scores
The mean of any distribution of Z scores is always 0. This is so because when you
change each raw score to a Z score, you take the raw score minus the mean. So the
mean is subtracted out of all the raw scores, making the overall mean come out to 0.
In other words, in any distribution, the sum of the positive Z scores must always equal
the sum of the negative Z scores. Thus, when you add them all up, you get 0.

The standard deviation of any distribution of Z scores is always 1. This is because
when you change each raw score to a Z score, you divide by the standard deviation.

A Z score is sometimes called a standard score. There are two reasons: Z scores
have standard values for the mean and the standard deviation, and, as we saw earlier,
Z scores provide a kind of standard scale of measurement for any variable. (However,
sometimes the term standard score is used only when the Z scores are for a distribu-
tion that follows a normal curve.)1

X = (Z) (SD) + M = (-1.73) (2.56) + 6.43 = -4.43 + 6.43 = 2.00

-1.73

How are you doing?

1. How is a Z score related to a raw score?
2. Write the formula for changing a raw score to a Z score, and define each of

the symbols.
3. For a particular group of scores, and . Give the Z score for

(a) 30, (b) 15, (c) 20, and (d) 22.5.
4. Write the formula for changing a Z score to a raw score, and define each of

the symbols.
5. For a particular group of scores, and . Give the raw score for

a Z score of (a) , (b) , (c) 0, and (d) .
6. Suppose a person has a Z score for overall health of and a Z score for

overall sense of humor of . What does it mean to say that this person is
healthier than she is funny?

+1
+2

-3+ .5+2
SD = 2M = 10

SD = 5M = 20

Answers

1.A Zscore is the number of standard deviations a raw score is above or below
the mean.

2.SD.Zis the Zscore; Xis the raw score; Mis the mean; SDis
the standard deviation.

3.(a)(b);(c)0;(d).5.
4.Xis the raw score; Zis the Zscore; SDis the standard de-

viation; Mis the mean.
5.(a)(b)11;(c)10;(d) 4.
6.This person is more above the average in health (in terms of how much people

typically vary from average in health) than she is above the average in humor
(in terms of how much people typically vary from the average in humor).

X=(Z)(SD)+M=(2)(2)+10=4+10=14;

X=(Z)(SD)+M.
-1 Z=(X-M)/SD=(30-20)/5=10/5=2;

Z=(X-M)/

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Some Key Ingredients for Inferential Statistics 73

Figure 3–6 A normal curve.

The Normal Curve
As noted in Chapter 1, the graphs of the distributions of many of the variables that
psychologists study follow a unimodal, roughly symmetrical, bell-shaped curve.
These bell-shaped smooth histograms approximate a precise and important mathe-
matical distribution called the normal distribution, or, more simply, the normal
curve.2 The normal curve is a mathematical (or theoretical) distribution. Re-
searchers often compare the actual distributions of the variables they are studying
(that is, the distributions they find in research studies) to the normal curve. They
don’t expect the distributions of their variables to match the normal curve perfectly
(since the normal curve is a theoretical distribution), but researchers often check
whether their variables approximately follow a normal curve. (The normal curve or
normal distribution is also often called a Gaussian distribution after the astronomer
Karl Friedrich Gauss. However, if its discovery can be attributed to anyone, it should
really be to Abraham de Moivre—see Box 3–1.) An example of the normal curve is
shown in Figure 3–6.

Why the Normal Curve Is So Common in Nature
Take, for example, the number of different letters a particular person can remem-
ber accurately on various testings (with different random letters each time). On
some testings the number of letters remembered may be high, on others low, and
on most somewhere in between. That is, the number of different letters a person
can recall on various testings probably approximately follows a normal curve.
Suppose that the person has a basic ability to recall, say, seven letters in this kind
of memory task. Nevertheless, on any particular testing, the actual number re-
called will be affected by various influences—noisiness of the room, the person’s
mood at the moment, a combination of random letters confused with a familiar
name, and so on.

These various influences add up to make the person recall more than seven on
some testings and less than seven on others. However, the particular combination of
such influences that come up at any testing is essentially random; thus, on most
testings, positive and negative influences should cancel out. The chances are not
very good of all the negative influences happening to come together on a testing
when none of the positive influences show up. Thus, in general, the person remem-
bers a middle amount, an amount in which all the opposing influences cancel each
other out. Very high or very low scores are much less common.

This creates a unimodal distribution with most of the scores near the middle
and fewer at the extremes. It also creates a distribution that is symmetrical, because
the number of letters recalled is as likely to be above as below the middle. Being a

normal distribution frequency distri-
bution that follows a normal curve.

normal curve specific, mathematically
defined, bell-shaped frequency distribu-
tion that is symmetrical and unimodal;
distributions observed in nature and in
research commonly approximate it.

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74 Chapter 3

In England, de Moivre was highly esteemed as a man
of letters as well as of numbers, being familiar with all
the classics and able to recite whole scenes from his
beloved Moliére’s Misanthropist. But for all his feelings
for his native France, the French Academy elected him a
foreign member of the Academy of Sciences just before
his death. In England, he was ineligible for a university
position because he was a foreigner there as well. He re-
mained in poverty, unable even to marry. In his earlier
years, he worked as a traveling teacher of mathematics.
Later, he was famous for his daily sittings in Slaughter’s
Coffee House in Long Acre, making himself available to
gamblers and insurance underwriters (two professions
equally uncertain and hazardous before statistics were
refined), who paid him a small sum for figuring odds for
them.

De Moivre’s unusual death generated several legends.
He worked a great deal with infinite series, which always
converge to a certain limit. One story has it that de
Moivre began sleeping 15 more minutes each night until
he was asleep all the time, then died. Another version
claims that his work at the coffeehouse drove him to such
despair that he simply went to sleep until he died. At any
rate, in his 80s he could stay awake only four hours a
day, although he was said to be as keenly intellectual in
those hours as ever. Then his wakefulness was reduced to
1 hour, then none at all. At the age of 87, after eight days
in bed, he failed to wake and was declared dead from
“somnolence” (sleepiness).

Sources: Pearson (1978); Tankard (1984).

BOX 3–1 de Moivre, the Eccentric Stranger Who Invented
the Normal Curve

The normal curve is central to statistics and is the foun-
dation of most statistical theories and procedures. If any
one person can be said to have discovered this fundamen-
tal of the field, it was Abraham de Moivre. He was a
French Protestant who came to England at the age of 21
because of religious persecution in France, which in 1685
denied Protestants all their civil liberties. In England, de
Moivre became a friend of Isaac Newton, who was sup-
posed to have often answered questions by saying, “Ask
Mr. de Moivre—he knows all that better than I do.” Yet
because he was a foreigner, de Moivre was never able to
rise to the same heights of fame as the British-born math-
ematicians who respected him so greatly.

Abraham de Moivre was mainly an expert on chance.
In 1733, he wrote a “method of approximating the sum
of the terms of the binomial expanded into a series.” His
paper essentially described the normal curve. The de-
scription was only in the form of a law, however; de
Moivre never actually drew the curve itself. In fact, he
was not very interested in it.

Credit for discovering the normal curve is often given
to Pierre Laplace, a Frenchman who stayed home; or Karl
Friedrich Gauss, a German; or Thomas Simpson, an Eng-
lishman. All worked on the problem of the distribution of
errors around a mean, even going so far as describing the
curve or drawing approximations of it. But even without
drawing it, de Moivre was the first to compute the areas
under the normal curve at 1, 2, and 3 standard deviations,
and Karl Pearson (discussed in Chapter 13, Box 13–1), a
distinguished later statistician, felt strongly that de Moivre
was the true discoverer of this important concept.

unimodal symmetrical curve does not guarantee that it will be a normal curve; it
could be too flat or too pointed. However, it can be shown mathematically that in the
long run, if the influences are truly random, and the number of different influences
being combined is large, a precise normal curve will result. Mathematical statisti-
cians call this principle the central limit theorem. We have more to say about this
principle in Chapter 5.

The Normal Curve and the Percentage of Scores Between
the Mean and 1 and 2 Standard Deviations from the Mean
The shape of the normal curve is standard. Thus, there is a known percentage of
scores above or below any particular point. For example, exactly 50% of the scores
in a normal curve are below the mean, because in any symmetrical distribution half

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Some Key Ingredients for Inferential Statistics 75

the scores are below the mean. More interestingly, as shown in Figure 3–7, approxi-
mately 34% of the scores are always between the mean and 1 standard deviation
from the mean.

Consider IQ scores. On many widely used intelligence tests, the mean IQ is 100,
the standard deviation is 16, and the distribution of IQs is roughly a normal curve
(see Figure 3–8). Knowing about the normal curve and the percentage of scores
between the mean and 1 standard deviation above the mean tells you that about 34%
of people have IQs between 100, the mean IQ, and 116, the IQ score that is 1 stan-
dard deviation above the mean. Similarly, because the normal curve is symmetrical,
about 34% of people have IQs between 100 and 84 (the score that is 1 standard devi-
ation below the mean), and 68% ( ) have IQs between 84 and 116.

There are many fewer scores between 1 and 2 standard deviations from the mean
than there are between the mean and 1 standard deviation from the mean. It turns out
that about 14% of the scores are between 1 and 2 standard deviations above the mean
(see Figure 3–7). (Similarly, about 14% of the scores are between 1 and 2 standard de-
viations below the mean.) Thus, about 14% of people have IQs between 116 (1 stan-
dard deviation above the mean) and 132 (2 standard deviations above the mean).

You will find it very useful to remember the 34% and 14% figures. These fig-
ures tell you the percentages of people above and below any particular score
whenever you know that score’s number of standard deviations above or below the
mean. You can also reverse this approach and figure out a person’s number of stan-
dard deviations from the mean from a percentage. Suppose you are told that a per-
son scored in the top 2% on a test. Assuming that scores on the test are
approximately normally distributed, the person must have a score that is at least 2
standard deviations above the mean. This is because a total of 50% of the scores
are above the mean, but 34% are between the mean and 1 standard deviation above

34% + 34%

−3 −2 −1 0
2%

14%
34% 34%

14%
2%

+1 +2 +3 Z Scores

Figure 3–7 Normal curve with approximate percentages of scores between the mean
and 1 and 2 standard deviations above and below the mean.

68 10084 116 132

IQ Scores

Figure 3–8 Distribution of IQ scores on many standard intelligence tests (with a mean
of 100 and a standard deviation of 16).

76 Chapter 3

the mean, and another 14% are between 1 and 2 standard deviations above the
mean. That leaves 2% of scores (that is, ) that are 2
standard deviations or more above the mean.

Similarly, suppose you were selecting animals for a study and needed to consider
their visual acuity. Suppose also that visual acuity was normally distributed and you
wanted to use animals in the middle two-thirds (a figure close to 68%) for visual
acuity. In this situation, you would select animals that scored between 1 standard
deviation above and 1 standard deviation below the mean. (That is, about 34% are
between the mean and 1 standard deviation above the mean and another 34% are be-
tween the mean and 1 standard deviation below the mean.) Also, remember that a Z
score is the number of standard deviations that a score is above or below the mean—
which is just what we are talking about here. Thus, if you knew the mean and the
standard deviation of the visual acuity test, you could figure out the raw scores (the
actual level of visual acuity) for being 1 standard deviation below and 1 standard de-
viation above the mean (that is, Z scores of and ). You would do this using the
methods of changing raw scores to Z scores and vice versa that you learned earlier in
this chapter, which are

The Normal Curve Table and Z Scores
The 50%, 34%, and 14% figures are important practical rules for working with a
group of scores that follow a normal distribution. However, in many research and ap-
plied situations, psychologists need more accurate information. Because the normal
curve is a precise mathematical curve, you can figure the exact percentage of scores
between any two points on the normal curve (not just those that happen to be right at
1 or 2 standard deviations from the mean). For example, exactly 68.59% of scores
have a Z score between and ; exactly 2.81% of scores have a Z score be-
tween and ; and so forth.

You can figure these percentages using calculus, based on the formula for the
normal curve. However, you can also do this much more simply (which you are
probably glad to know!). Statisticians have worked out tables for the normal curve
that give the percentage of scores between the mean (a Z score of 0) and any other Z
score (as well as the percentage of scores in the tail for any Z score).

Wehave includedanormalcurve table in theAppendix (TableA–1,pp.664–667).
Table 3–1 shows the first part of the full table. The first column in the table lists the
Z score. The second column, labeled “% Mean to Z,” gives the percentage of scores
between the mean and that Z score. The shaded area in the curve at the top of the col-
umn gives a visual reminder of the meaning of the percentages in the column. The
third column, labeled “% in Tail,” gives the percentage of scores in the tail for that Z
score. The shaded tail area in the curve at the top of the column shows the meaning
of the percentages in the column. Notice that the table lists only positive Z scores.
This is because the normal curve is perfectly symmetrical. Thus, the percentage of
scores between the mean and, say, a Z of (which is 33.65%) is exactly the same
as the percentage of scores between the mean and a Z of (again 33.65%); and
the percentage of scores in the tail for a Z score of (3.84%) is the same as the
percentage of scores in the tail for a Z score of (again, 3.84%). Notice that for
each Z score, the “% Mean to Z ” value and the “% in Tail” value sum to 50.00. This
is because exactly 50% of the scores are above the mean for a normal curve. For ex-
ample, for the Z score of .57, the “% Mean to Z” value is 21.57% and the “% in Tail”
value is 28.43%, and .

Suppose you want to know the percentage of scores between the mean and a
Z score of .64. You just look up .64 in the “Z” column of the table and the “% Mean

21.57% + 28.43% = 50.00%

-1.77
+1.77

– .98
+ .98

+ .89+ .79
-1.68+ .62

Z = (X – M)/SD and

X = (Z)(SD) + M.

+1-1

50% – 34% – 14% = 2%

T I P F O R S U C C E S S
Remember that negative Z scores
are scores below the mean and
positive Z scores are scores above
the mean.

normal curve table table showing
percentages of scores associated with the
normal curve; the table usually includes
percentages of scores between the mean
and various numbers of standard devia-
tions above the mean and percentages of
scores more positive than various num-
bers of standard deviations above the
mean.

Some Key Ingredients for Inferential Statistics 77

to Z” column tells you that 23.89% of the scores in a normal curve are between the
mean and this Z score. These values are highlighted in Table 3–1.

You can also reverse the process and use the table to find the Z score for a par-
ticular percentage of scores. For example, imagine that 30% of ninth-grade students
had a creativity score higher than Janice’s. Assuming that creativity scores follow a
normal curve, you can figure out her Z score as follows: if 30% of students scored
higher than she did, then 30% of the scores are in the tail above her score. This is
shown in Figure 3–9. So, you would look at the “% in Tail” column of the table until
you found the percentage that was closest to 30%. In this example, the closest is
30.15%. Finally, look at the “Z” column to the left of this percentage, which lists a Z
score of .52 (these values of 30.15% and .52 are highlighted in Table 3–1). Thus,
Janice’s Z score for her level of creativity is .52. If you know the mean and standard
deviation for ninth-grade students’ creativity scores, you can figure out Janice’s ac-
tual raw score on the test by changing her Z score of .52 to a raw score using the
usual formula, .X = (Z)(SD) + (M)

T I P F O R S U C C E S S
Notice that the table repeats the
basic three columns twice on the
page. Be sure to look across to
the columns you need.

Table 3–1 Normal Curve Areas: Percentage of the Normal Curve Between the Mean and the
Scores Shown and Percentage of Scores in the Tail for the Z Scores Shown (First
part of table only: full table is Table A–1 in the Appendix. Highlighted values are
examples from the text.)

Z % Mean to Z % in Tail Z % Mean to Z % in Tail

.00 .00 50.00 .45 17.36 32.64

.01 .40 49.60 .46 17.72 32.28

.02 .80 49.20 .47 18.08 31.92

.03 1.20 48.80 .48 18.44 31.56

.04 1.60 48.40 .49 18.79 31.21

.05 1.99 48.01 .50 19.15 30.85

.06 2.39 47.61 .51 19.50 30.50

.07 2.79 47.21 .52 19.85 30.15

.08 3.19 46.81 .53 20.19 29.81

.09 3.59 46.41 .54 20.54 29.46

.10 3.98 46.02 .55 20.88 29.12

.11 4.38 45.62 .56 21.23 28.77

.12 4.78 45.22 .57 21.57 28.43

.13 5.17 44.83 .58 21.90 28.10

.14 5.57 44.43 .59 22.24 27.76

.15 5.96 44.04 .60 22.57 27.43

.16 6.36 43.64 .61 22.91 27.09

.17 6.75 43.25 .62 23.24 26.76

.18 7.14 42.86 .63 23.57 26.43

.19 7.53 42.47 .64 23.89 26.11

.20 7.93 42.07 .65 24.22 25.78

.21 8.32 41.68 .66 24.54 25.46

mean Zmean Zmean Zmean Z

78 Chapter 3

Steps for Figuring the Percentage of Scores Above
or Below a Particular Raw Score or Z Score Using
the Normal Curve Table
Here are the five steps for figuring the percentage of scores.

❶ If you are beginning with a raw score, first change it to a Z score. Use the
usual formula, .

❷ Draw a picture of the normal curve, where the Z score falls on it, and shade
in the area for which you are finding the percentage. (When marking where
the Z score falls on the normal curve, be sure to put it in the right place above or
below the mean according to whether it is a positive or negative Z score.)

❸ Make a rough estimate of the shaded area’s percentage based on the
50%–34%–14% percentages. You don’t need to be very exact; it is enough
just to estimate a range in which the shaded area has to fall, figuring it is be-
tween two particular whole Z scores. This rough estimate step is designed not
only to help you avoid errors (by providing a check for your figuring), but also
to help you develop an intuitive sense of how the normal curve works.

❹ Find the exact percentage using the normal curve table, adding 50% if nec-
essary. Look up the Z score in the “Z” column of Table A–1 and find the percent-
age in the “% Mean to Z” column or “% in Tail” column next to it. If you want
the percentage of scores between the mean and this Z score, or if you want the
percentage of scores in the tail for this Z score, the percentage in the table is your
final answer. However, sometimes you need to add 50% to the percentage in the
table. You need to do this if the Z score is positive and you want the total percent-
age below this Z score, or if the Z score is negative and you want the total per-
centage above this Z score. However, you don’t need to memorize these rules; it
is much easier to make a picture for the problem and reason out whether the per-
centage you have from the table is correct as is or if you need to add 50%.

❺ Check that your exact percentage is within the range of your rough esti-
mate from Step ❸.

Examples
Here are two examples using IQ scores where and .

Example 1: If a person has an IQ of 125, what percentage of people have higher
IQs?

SD = 16M = 100

Z = (X – M)/SD

50% 30%

.52 1 2

Figure 3–9 Distribution of creativity test scores showing area for top 30% of scores
and Z score where this area begins.

Some Key Ingredients for Inferential Statistics 79

❶ If you are beginning with a raw score, first change it to a Z score. Using the
usual formula, .

❷ Draw a picture of the normal curve, where the Z score falls on it, and shade
in the area for which you are finding the percentage. This is shown in
Figure 3–10 (along with the exact percentages figured later).

❸ Make a rough estimate of the shaded area’s percentage based on the
50%–34%–14% percentages. If the shaded area started at a Z score of 1, it
would have 16% above it. If it started at a Z score of 2, it would have only 2%
above it. So, with a Z score of 1.56, the number of scores above it has to be
somewhere between 16% and 2%.

❹ Find the exact percentage using the normal curve table, adding 50% if nec-
essary. In Table A–1, 1.56 in the “Z” column goes with 5.94 in the “% in Tail”
column. Thus, 5.94% of people have IQ scores higher than 125. This is the an-
swer to our problem. (There is no need to add 50% to the percentage.)

❺ Check that your exact percentage is within the range of your rough estimate
from Step ❸. Our result, 5.94%, is within the 16-to-2% range we estimated.

Example 2: If a person has an IQ of 95, what percentage of people have higher
IQs?

❶ If you are beginning with a raw score, first change it to a Z score. Using the
usual formula, .

❷ Draw a picture of the normal curve, where the Z score falls on it, and
shade in the area for which you are finding the percentage. This is shown in
Figure 3–11 (along with the percentages figured later).

Z = (95 – 100)/16 =

– .31

Z = (X – M)/SD, Z = (125 – 100)/16 = +1.56

68 10084 116 132125

Z Score:

IQ Score:

0−1 +2+1.56+1−2

5.94%

50%

Figure 3–10 Distribution of IQ scores showing percentage of scores above an IQ
score of 125 (shaded area).

68 10084 116 132
Z Score:
IQ Score:

0- .31 +2+1−2

62.1

95
−1

12
.1

7%
50%

Figure 3–11 Distribution of IQ scores showing percentage of scores above an IQ score
of 95 (shaded area).

80 Chapter 3

❸ Make a rough estimate of the shaded area’s percentage based on the 50%–
34%–14% percentages. You know that 34% of the scores are between the
mean and a Z score of . Also, 50% of the curve is above the mean. Thus, the
Z score of has to have between 50% and 84% of scores above it.

❹ Find the exact percentage using the normal curve table, adding 50% if nec-
essary. The table shows that 12.17% of scores are between the mean and a Z
score of .31. Thus, the percentage of scores above a Z score of is the
12.17% between the Z score and the mean plus the 50% above the mean, which
is 62.17%.

❺ Check that your exact percentage is within the range of your rough esti-
mate from Step ❸. Our result of 62.17% is within the 50-to-84% range we
estimated.

Figuring Z Scores and Raw Scores from Percentages
Using the Normal Curve Table
Going from a percentage to a Z score or raw score is similar to going from a Z score
or raw score to a percentage. However, you reverse the procedure when figuring the
exact percentage. Also, any necessary changes from a Z score to a raw score are done
at the end.

Here are the five steps.

❶ Draw a picture of the normal curve, and shade in the approximate area for
your percentage using the 50%–34%–14% percentages.

❷ Make a rough estimate of the Z score where the shaded area stops.
❸ Find the exact Z score using the normal curve table (subtracting 50% from

your percentage if necessary before looking up the Z score). Looking at your
picture, figure out either the percentage in the shaded tail or the percentage be-
tween the mean and where the shading stops. For example, if your percentage is
the bottom 35%, then the percentage in the shaded tail is 35%. Figuring the per-
centage between the mean and where the shading stops will sometimes involve
subtracting 50% from the percentage in the problem. For example, if your per-
centage is the top 72%, then the percentage from the mean to where that shading
stops is 22% ( ).

Once you have the percentage, look up the closest percentage in the appro-
priate column of the normal curve table (“% Mean to Z” or “% in Tail”) and find
the Z score for that percentage. That Z will be your answer—except it may be
negative. The best way to tell if it is positive or negative is by looking at your
picture.

❹ Check that your exact Z score is within the range of your rough estimate
from Step ❷.

❺ If you want to find a raw score, change it from the Z score. Use the usual for-
mula,

Examples
Here are three examples. Once again, we use IQ for our examples, with
and .

Example 1: What IQ score would a person need to be in the top 5%?

❶ Draw a picture of the normal curve, and shade in the approximate area for
your percentage using the 50%–34%–14% percentages. We wanted the top
5%. Thus, the shading has to begin above (to the right of) 1 SD (there are 16%

SD = 16
M = 100

X = (Z)(SD) + M.

72% – 50% = 22%

– .31

– .31
-1

Some Key Ingredients for Inferential Statistics 81

of scores above 1 SD). However, it cannot start above 2 SD because only 2% of
all the scores are above 2 SD. But 5% is a lot closer to 2% than to 16%. Thus,
you would start shading a small way to the left of the 2 SD point. This is shown
in Figure 3–12.

❷ Make a rough estimate of the Z score where the shaded area stops. The Z
score is between and .

❸ Find the exact Z score using the normal curve table (subtracting 50% from
your percentage if necessary before looking up the Z score). We want the top
5%; so we can use the “% in Tail” column of the normal curve table. Looking in
that column, the closest percentage to 5% is 5.05% (or you could use 4.95%).
This goes with a Z score of 1.64 in the “Z” column.

❹ Check that your exact Z score is within the range of your rough estimate
from Step ❷. As we estimated, is between and (and closer to 2).

❺ If you want to find a raw score, change it from the Z score. Using the formula,
. In sum, to be in the top

5%, a person would need an IQ of at least 126.24.

Example 2: What IQ score would a person need to be in the top 55%?

❶ Draw a picture of the normal curve and shade in the approximate area for
your percentage using the 50%–34%–14% percentages. You want the top
55%. There are 50% of scores above the mean. So, the shading has to begin
below (to the left of) the mean. There are 34% of scores between the mean and
1 SD below the mean; so the score is between the mean and 1 SD below the
mean. You would shade the area to the right of that point. This is shown in
Figure 3–13.

X = (Z)(SD) + M = (1.64)(16) + 100 = 126.24

+2+1+1.64

+2+1

68 10084 116 132126.24

Z Score:
IQ Score:

0−1 +2+1.64+1−2

50%
5%

Figure 3–12 Finding the Z score and IQ raw score for where the top 5% of scores
start.

68 10084 116 13297.92

Z Score:
IQ Score:

0−1 +2−.13 +1−2

50%
5%

55%

Figure 3–13 Finding the IQ score for where the top 55% of scores start.

82 Chapter 3

❷ Make a rough estimate of the Z score where the shaded area stops. The Z
score has to be between 0 and .

❸ Find the exact Z score using the normal curve table (subtracting 50% from
your percentage if necessary before looking up the Z score). Being in the top
55% means that 5% of people have IQs between this IQ and the mean (that is,

). In the normal curve table, the closest percentage to 5% in
the “% Mean to Z” column is 5.17%, which goes with a Z score of .13. Because
you are below the mean, this becomes .

❹ Check that your exact Z score is within the range of your rough estimate
from Step ❷. As we estimated, is between 0 and .

➎ If you want to find a raw score, change it from the Z score. Using the usual
formula, . So, to be in the top 55% on IQ, a per-
son needs an IQ score of 97.92 or higher.

Example 3: What range of IQ scores includes the 95% of people in the middle
range of IQ scores?
This kind of problem—finding the middle percentage—may seem odd. How-

ever, it is actually a very common situation used in procedures you will learn in later
chapters.

Think of this kind of problem in terms of finding the scores that go with the
upper and lower ends of this percentage. Thus, in this example, you are trying to find
the points where the bottom 2.5% ends and the top 2.5% begins (which, out of
100%, leaves the middle 95%).

❶ Draw a picture of the normal curve, and shade in the approximate area for
your percentage using the 50%–34%–14% percentages. Let’s start where
the top 2.5% begins. This point has to be higher than 1 SD (16% of scores are
higher than 1 SD). However, it cannot start above 2 SD because there are only
2% of scores above 2 SD. But 2.5% is very close to 2%. Thus, the top 2.5%
starts just to the left of the 2 SD point. Similarly, the point where the bottom
2.5% comes in is just to the right of SD. The result of all this is that we will
shade in two tail areas on the curve: one starting just above SD and the other
starting just below SD. This is shown in Figure 3–14.

❷ Make a rough estimate of the Z score where the shaded area stops. You can
see from the picture that the Z score for where the shaded area stops above the
mean is just below . Similarly, the Z score for where the shaded area stops
below the mean is just above .

❸ Find the exact Z score using the normal curve table (subtracting 50% from
your percentage if necessary before looking up the Z score). Being in the top
2.5% means that 2.5% of the IQ scores are in the upper tail. In the normal curve
table, the closest percentage to 2.5% in the “% in Tail” column is exactly 2.50%,

-2

+2
-2

-2

X = (- .13)(16) + 100 = 97.92

-1

– .13

55% – 50% = 5%

-1
68 10084 116 132
Z Score:
IQ Score:

0−1 +2+1−2

95%

−1.96 +1.96

2.5% 2.5%

Figure 3–14 Finding the IQ scores for where the middle 95% of scores begins and ends.

Some Key Ingredients for Inferential Statistics 83

which goes with a Z score of . The normal curve is symmetrical. Thus, the
Z score for the lower tail is .

❹ Check that your exact Z score is within the range of your rough estimate
from Step ❷. As we estimated, is between and and is very close
to , and is between and and very close to .

❺ If you want to find a raw score, change it from the Z score. For the high
end, using the usual formula, . For the low end,

. In sum, the middle 95% of IQ scores run
from 68.64 to 131.36.
X = (-1.96)(16) + 100 = 68.64

X = (1.96)(16) + 100 = 131.36

-2-2-1-1.96+2
+2+1+1.96

-1.96
+1.96

How are you doing?

1. Why is the normal curve (or at least a curve that is symmetrical and unimodal)
so common in nature?

2. Without using a normal curve table, about what percentage of scores on a
normal curve are (a) above the mean, (b) between the mean and 1 SD above
the mean, (c) between 1 and 2 SDs above the mean, (d) below the mean, (e)
between the mean and 1 SD below the mean, and (f) between 1 and 2 SDs
below the mean?

3. Without using a normal curve table, about what percentage of scores on a
normal curve are (a) between the mean and 2 SDs above the mean, (b) below
1 SD above the mean, (c) above 2 SDs below the mean?

4. Without using a normal curve table, about what Z score would a person have
who is at the start of the top (a) 50%, (b) 16%, (c) 84%, (d) 2%?

5. Using the normal curve table, what percentage of scores are (a) between the
mean and a Z score of 2.14, (b) above 2.14, (c) below 2.14?

6. Using the normal curve table, what Z score would you have if (a) 20% are
above you and (b) 80% are below you?

Answers

1.It is common because any particular score is the result of the random combi-
nation of many effects, some of which make the score larger and some of
which make the score smaller. Thus, on average these effects balance out near
the middle, with relatively few at each extreme, because it is unlikely for most
of the increasing and decreasing effects to come out in the same direction.

2.(a)Above the mean: 50%;(b)between the mean and 1 SDabove the mean:
34%;(c)between 1 and 2 SDs above the mean: 14%;(d)below the mean:
50%;(e)between the mean and 1 SDbelow the mean: 34%;(f)between 1
and 2 SDs below the mean: 14%.

3.(a)Between the mean and 2 SDs above the mean: 48%;(b)below 1 SD
above the mean: 84%;(c)above 2 SDs below the mean: 98%.

4.(a)50%: 0;(b)16%: 1;(c)84%: ;(d)2%: 2.
5.(a)Between the mean and a Zscore of 2.14: 48.38%;(b)above 2.14: 1.62%;

(c)below 2.14: 98.38%.
6.(a)20% above you: .84;(b)80% below you: .84.

-1

Sample and Population
We are going to introduce you to some important ideas by thinking of beans. Sup-
pose you are cooking a pot of beans and taste a spoonful to see if they are done.
In this example, the pot of beans is a population, the entire set of things of interest.
The spoonful is a sample, the part of the population about which you actually have

population entire group of people to
which a researcher intends the results of
a study to apply; larger group to which
inferences are made on the basis of the
particular set of people (sample) studied.

sample scores of the particular group
of people studied; usually considered to
be representative of the scores in some
larger population.

84 Chapter 3

information. This is shown in Figure 3–15a. Figures 3–15b and 3–15c are other ways
of showing the relation of a sample to a population.

In psychology research, we typically study samples not of beans but of individ-
uals to make inferences about some larger group (a population). A sample might con-
sist of the scores of 50 Canadian women who participate in a particular experiment,
whereas the population might be intended to be the scores of all Canadian women. In
an opinion survey, 1,000 people might be selected from the voting-age population of
a particular district and asked for whom they plan to vote. The opinions of these
1,000 people are the sample. The opinions of the larger voting public in that country,
to which the pollsters apply their results, is the population (see Figure 3–16).

Why Psychologists Study Samples Instead of Populations
If you want to know something about a population, your results would be most accu-
rate if you could study the entire population rather than a subgroup from it. However,
in most research situations this is not practical. More important, the whole point of
research usually is to be able to make generalizations or predictions about events be-
yond your reach. We would not call it scientific research if we tested three particular
cars to see which gets better gas mileage—unless you hoped to say something about
the gas mileage of those models of cars in general. In other words, a researcher
might do an experiment on how people store words in short-term memory using
20 students as the participants in the experiment. But the purpose of the experiment
is not to find out how these particular 20 students respond to the experimental versus
the control condition. Rather, the purpose is to learn something about human memory
under these conditions in general.

The strategy in almost all psychology research is to study a sample of individu-
als who are believed to be representative of the general population (or of some par-
ticular population of interest). More realistically, researchers try to study people who
do not differ from the general population in any systematic way that should matter
for that topic of research.

The sample is what is studied, and the population is an unknown about which
researchers draw conclusions based on the sample. Most of what you learn in the rest
of this book is about the important work of drawing conclusions about populations
based on information from samples.

(a) (b) (c)

Figure 3–15 Populations and samples: (a) The entire pot of beans is the population,
and the spoonful is the sample. (b) The entire larger circle is the population, and the circle
within it is the sample. (c) The histogram is of the population, and the particular shaded scores
make up the sample.

Some Key Ingredients for Inferential Statistics 85

Methods of Sampling
Usually, the ideal method of picking out a sample to study is called random selec-
tion. The researcher starts with a complete list of the population and randomly se-
lects some of them to study. An example of random selection is to put each name
on a table tennis ball, put all the balls into a big hopper, shake it up, and have a
blindfolded person select as many as are needed. (In practice, most researchers use
a computer-generated list of random numbers. Just how computers or persons can
create a list of truly random numbers is an interesting question in its own right that
we examine in Chapter 14, Box 14–1.)

It is important not to confuse truly random selection with what might be called
haphazard selection; for example, just taking whoever is available or happens
to be first on a list. When using haphazard selection, it is surprisingly easy to pick

All
Canadian
Women

50
Canadian
Women

All
Voters

1,000
Voters

(a)

(b)

Figure 3–16 Additional examples of populations and samples: (a) The population is
the scores of all Canadian women, and the sample is the scores of the 50 Canadian women
studied. (b) The population is the voting preferences of the entire voting-age population, and
the sample is the voting preferences of the 1,000 voting-age people who were surveyed.

random selection method for select-
ing a sample that uses truly random pro-
cedures (usually meaning that each
person in the population has an equal
chance of being selected); one procedure
is for the researcher to begin with a com-
plete list of all the people in the popula-
tion and select a group of them to study
using a table of random numbers.

86 Chapter 3

accidentally a group of people that is really quite different from the population as a
whole. Consider a survey of attitudes about your statistics instructor. Suppose you
give your questionnaire only to other students sitting near you in class. Such a sur-
vey would be affected by all the things that influence where students choose to sit,
some of which have to do with the topic of your study—how much students like the
instructor or the class. Thus, asking students who sit near you would likely result in
opinions more like your own than a truly random sample would.

Unfortunately, it is often impractical or impossible to study a truly random sam-
ple. Much of the time, in fact, studies are conducted with whoever is willing or avail-
able to be a research participant. At best, as noted, a researcher tries to study a
sample that is not systematically unrepresentative of the population in any known
way. For example, suppose a study is about a process that is likely to differ for peo-
ple of different age groups. In this situation, the researcher may attempt to include
people of all age groups in the study. Alternatively, the researcher would be careful
to draw conclusions only about the age group studied.

Methods of sampling is a complex topic that is discussed in detail in research
methods textbooks (also see Box 3–2) and in the research methods Web Chapter W1
(Overview of the Logic and Language of Psychology Research) on the Web site for
this book http://www.pearsonhighered.com/.

better houses and better neighborhoods. In 1948, the
election was very close, and the Republican bias pro-
duced the embarrassing mistake that changed survey
methods forever.

Since 1948, all survey organizations have used what
is called a “probability method.” Simple random sam-
pling is the purest case of the probability method, but
simple random sampling for a survey about a U.S. presi-
dential election would require drawing names from a list
of all the eligible voters in the nation—a lot of people.
Each person selected would have to be found, in diversely
scattered locales. So instead, “multistage cluster sam-
pling” is used. The United States is divided into seven
size-of-community groupings, from large cities to rural
open country; these groupings are divided into seven
geographic regions (New England, Middle Atlantic, and
so on), after which smaller equal-sized groups are zoned,
and then city blocks are drawn from the zones, with the
probability of selection being proportional to the size of
the population or number of dwelling units. Finally, an
interviewer is given a randomly selected starting point
on the map and is required to follow a given direction,
taking households in sequence.

Actually, telephoning is often the favored method for
polling today. Phone surveys cost about one-third of
door-to-door polls. Since most people now own phones,
this method is less biased than in Truman’s time. Phoning

BOX 3–2 Surveys, Polls, and 1948’s Costly “Free Sample”
It is time to make you a more informed reader of polls in
the media. Usually the results of properly done public
polls are accompanied, somewhere in fine print, by a
statement such as, “From a telephone poll of 1,000
American adults taken on June 4 and 5. Sampling error

.” What does a statement like this mean?
The Gallup poll is as good an example as any (Gallup,

1972; see also http://www.gallup.com), and there is no
better place to begin than in 1948, when all three of the
major polling organizations—Gallup, Crossley (for
Hearst papers), and Roper (for Fortune)—wrongly pre-
dicted Thomas Dewey’s victory over Harry Truman for
the U.S. presidency. Yet Gallup’s prediction was based
on 50,000 interviews and Roper’s on 15,000. By con-
trast, to predict George H. W. Bush’s 1988 victory,
Gallup used only 4,089. Since 1952, the pollsters have
never used more than 8,144—but with very small error
and no outright mistakes. What has changed?

The method used before 1948, and never repeated
since, was called “quota sampling.” Interviewers were
assigned a fixed number of persons to interview, with
strict quotas to fill in all the categories that seemed im-
portant, such as residence, sex, age, race, and economic
status. Within these specifics, however, they were free to
interview whomever they liked. Republicans generally
tended to be easier to interview. They were more likely to
have telephones and permanent addresses and to live in

;3%

Some Key Ingredients for Inferential Statistics 87

Statistical Terminology for Samples and Populations
The mean, variance, and standard deviation of a population are called population pa-
rameters. A population parameter usually is unknown and can be estimated only from
what you know about a sample taken from that population. You do not taste all the
beans, just the spoonful. “The beans are done” is an inference about the whole pot.

Population parameters are usually shown as Greek letters (e.g., ). (This is a
statistical convention with origins tracing back more than 2,000 years to the early
Greek mathematicians.) The symbol for the mean of a population is , the Greek let-
ter mu. The symbol for the variance of a population is , and the symbol for its stan-
dard deviation is , the lowercase Greek letter sigma. You won’t see these symbols
often, except while learning statistics. This is because, again, researchers seldom
know the population parameters.

The mean, variance, and standard deviation you figure for the scores in a sample
are called sample statistics. A sample statistic is figured from known information.
Sample statistics are what we have been figuring all along and are expressed with the
roman letters you learned in Chapter 2: M, , and SD. The population parameter
and sample statistic symbols for the mean, variance, and standard deviation are sum-
marized in Table 3–2.

The use of different types of symbols for population parameters (Greek letters)
and sample statistics (roman letters) can take some getting used to; so don’t worry if
it seems tricky at first. It’s important to know that the statistical concepts you are

SD2

�

�2

�
�

also allows computers to randomly dial phone numbers
and, unlike telephone directories, this method calls unlist-
ed numbers. However, survey organizations in the United
States typically do not call cell phone numbers. Thus,
U.S. households that use a cell phone for all calls and do
not have a home phone are not usually included in tele-
phone opinion polls. Most survey organizations consider
the current cell-phone-only rate to be low enough not to
cause large biases in poll results (especially since the de-
mographic characteristics of individuals without a home
phone suggest that they are less likely to vote than indi-
viduals who live in households with a home phone).
However, anticipated future increases in the cell-phone-
only rate will likely make this an important issue for opin-
ion polls. Survey organizations will need to consider

additional polling methods, perhaps using the Internet
and email.

Whether by telephone or face to face, there will be
about 35% nonrespondents after three attempts. This cre-
ates yet another bias, dealt with through questions about
how much time a person spends at home, so that a slight
extra weight can be given to the responses of those
reached but usually at home less, to make up for those
missed entirely.

Now you know quite a bit about opinion polls, but we
have left two important questions unanswered: Why are
only about 1,000 included in a poll meant to describe all
U.S. adults, and what does the term sampling error
mean? For these answers, you must wait for Chapter 5
(Box 5–1).

Table 3–2 Population Parameters and Sample Statistics

Population Parameter
(Usually Unknown)

Sample Statistic
(Figured from Known Data)

Basis: Scores of entire population Scores of sample only

Symbols:

Mean M

Standard deviation SD

Variance SD 2�2
�

�

population mean.�

population variance.�2

population standard deviation.�

sample statistics descriptive statistic,
such as the mean or standard deviation,
figured from the scores in a group of
people studied.

population parameter actual value of
the mean, standard deviation, and so on,
for the population; usually population
parameters are not known, though often
they are estimated based on information
in samples.

88 Chapter 3

learning—such as the mean, variance, and standard deviation—are the same for both
a population and a sample. So, for example, you have learned that the standard devi-
ation provides a measure of the variability of the scores in a distribution—whether
we are talking about a sample or a population. (You will learn in later chapters that
the variance and standard deviation are figured in a different way for a population
than for a sample, but the concepts do not change). We use different symbols for
population parameters and sample statistics to make it clear whether we are referring
to a population or a sample. This is important, because some of the formulas you will
encounter in later chapters use both sample statistics and population parameters.

How are you doing?

1. Explain the difference between the population and a sample for a research
study.

2. Why do psychologists usually study samples and not populations?
3. Explain the difference between random sampling and haphazard sampling.
4. Explain the difference between a population parameter and a sample statistic.
5. Give the symbols for the population parameters for (a) the mean and (b) the

standard deviation.
6. Why are different symbols (Greek versus roman letters) used for population

parameters and sample statistics?

Answers

1.The population is the entire group to which results of a study are intended to
apply. The sample is the particular, smaller group of individuals actually studied.

2.Psychologists usually study samples and not populations because it is not
practical in most cases to study the entire population.

3.In random sampling, the sample is chosen from among the population using
a completely random method, so that each individual has an equal chance of
being included in the sample. In haphazard sampling, the researcher selects
individuals who are easily available or who are convenient to study.

4.A population parameter is about the population (such as the mean of all the
scores in the population); a sample statistic is about a particular sample (such
as the mean of the scores of the people in the sample).

5.(a)Mean: ;(b) standard deviation: .
6.Using different symbols for population parameters and sample statistics en-

sures that there is no confusion as to whether a symbol refers to a population
or a sample.

� �

Probability
The purpose of most psychological research is to examine the truth of a theory or the
effectiveness of a procedure. But scientific research of any kind can only make that
truth or effectiveness seem more or less likely; it cannot give us the luxury of know-
ing for certain. Probability is very important in science. In particular, probability is
very important in inferential statistics, the methods psychologists use to go from re-
sults of research studies to conclusions about theories or applied procedures.

Probability has been studied for centuries by mathematicians and philosophers.
Yet even today the topic is full of controversy. Fortunately, however, you need to
know only a few key ideas to understand and carry out the inferential statistical pro-
cedures you learn in this book. These few key points are not very difficult; indeed,
some students find them to be quite intuitive.

Some Key Ingredients for Inferential Statistics 89

Interpretations of Probability
In statistics, we usually define probability as the expected relative frequency of a
particular outcome. An outcome is the result of an experiment (or just about any sit-
uation in which the result is not known in advance, such as a coin coming up heads
or it raining tomorrow). Frequency is how many times something happens. The
relative frequency is the number of times something happens relative to the number
of times it could have happened; that is, relative frequency is the proportion of times
something happens. (A coin might come up heads 8 times out of 12 flips, for a rela-
tive frequency of 8�12, or 2�3.) Expected relative frequency is what you expect to
get in the long run if you repeat the experiment many times. (In the case of a coin, in
the long run you would expect to get 1�2 heads). This is called the long-run relative-
frequency interpretation of probability.

We also use probability to express how certain we are that a particular thing will
happen. This is called the subjective interpretation of probability. Suppose that
you say there is a 95% chance that your favorite restaurant will be open tonight. You
could be using a kind of relative frequency interpretation. This would imply that if
you were to check whether this restaurant was open many times on days like today,
you would find it open on 95% of those days. However, what you mean is probably
more subjective: on a scale of 0% to 100%, you would rate your confidence that the
restaurant is open at 95%. To put it another way, you would feel that a fair bet would
have odds based on a 95% chance of the restaurant’s being open.

The interpretation, however, does not affect how probability is figured. We men-
tion these interpretations because we want to give you a deeper insight into the mean-
ing of the term probability, which is such a prominent concept throughout statistics.

Figuring Probabilities
Probabilities are usually figured as the proportion of successful possible outcomes—
the number of possible successful outcomes divided by the number of all possible
outcomes. That is,

Consider the probability of getting heads when flipping a coin. There is one possi-
ble successful outcome (getting heads) out of two possible outcomes (getting heads or
getting tails). This makes a probability of 1�2, or .5. In a throw of a single die, the
probability of a 2 (or any other particular side of the six-sided die) is 1�6, or .17. This
is because there can be only one successful outcome out of six possible outcomes. The
probability of throwing a die and getting a number 3 or lower is 3�6, or .5. There are
three possible successful outcomes (a 1, a 2, or a 3) out of six possible outcomes.

Probability =
Possible successful outcomes

All possible outcomes

probability expected relative frequency
of an outcome; the proportion of suc-
cessful outcomes to all outcomes.

outcome term used in discussing
probability for the result of an experi-
ment (or almost any event, such as a
coin coming up heads or it raining
tomorrow).

expected relative frequency number
of successful outcomes divided by the
number of total outcomes you would ex-
pect to get if you repeated an experiment
a large number of times.

long-run relative-frequency interpre-
tation of probability understanding
of probability as the proportion of a par-
ticular outcome that you would get if the
experiment were repeated many times.

subjective interpretation of probabil-
ity way of understanding probability as
the degree of one’s certainty that a par-
ticular outcome will occur.

planned, how much of the stakes should each player walk
away with, given the percentage of plays completed?

The problem was discussed at least as early as 1494
by Luca Pacioli, a friend of Leonardo da Vinci. But it
was unsolved until 1654, when it was presented to Blaise
Pascal by the Chevalier de Méré. Pascal, a French child

BOX 3–3 Pascal Begins Probability Theory at the Gambling Table,
Then Learns to Bet on God

Whereas in England, statistics were used to keep track of
death rates and to prove the existence of God (see Chapter 1,
Box 1–1), the French and Italians developed statistics at
the gaming table. In particular, there was the “problem of
points”—the division of the stakes in a game after it has
been interrupted. If a certain number of plays were

90 Chapter 3

Now consider a slightly more complicated example. Suppose a class has 200
people in it, and 30 are seniors. If you were to pick someone from the class at random,
the probability of picking a senior would be 30�200, or .15. This is because there are
30 possible successful outcomes (getting a senior) out of 200 possible outcomes.

Steps for Finding Probabilities
There are three steps for finding probabilities.

❶ Determine the number of possible successful outcomes.
❷ Determine the number of all possible outcomes.
❸ Divide the number of possible successful outcomes (Step ❶) by the number

of all possible outcomes (Step ❷).

Let’s apply these steps to the probability of getting a number 3 or lower on a
throw of a die.

❶ Determine the number of possible successful outcomes. There are three out-
comes of 3 or lower: 1, 2, or 3.

❷ Determine the number of all possible outcomes. There are six possible out-
comes in the throw of a die: 1, 2, 3, 4, 5, or 6.

❸ Divide the number of possible successful outcomes (Step ❶) by the number
of all possible outcomes (Step ❷). .

Range of Probabilities
A probability is a proportion, the number of possible successful outcomes to the total
number of possible outcomes. A proportion cannot be less than 0 or greater than 1. In
terms of percentages, proportions range from 0% to 100%. Something that has no
chance of happening has a probability of 0, and something that is certain to happen
has a probability of 1. Notice that when the probability of an event is 0, the event is
completely impossible; it cannot happen. But when the probability of an event is
low, say 5% or even 1%, the event is improbable or unlikely, but not impossible.

Probabilities Expressed as Symbols
Probability is usually symbolized by the letter p. The actual probability number is
usually given as a decimal, though sometimes fractions or percentages are used. A
50-50 chance is usually written as , but it could also be written as orp = 1>2p = .5

3>6 = .5

prodigy, attended meetings of the most famous adult
French mathematicians and at 15 proved an important
theorem in geometry. In correspondence with Pierre de
Fermat, another famous French mathematician, Pascal
solved the problem of points and in so doing began the
field of probability theory and the work that would lead
to the normal curve. (For more information on the prob-
lem of points, including its solution, see http:/ /mathforum.
org/isaac/problems/prob1.html).

Not long after solving this problem, Pascal became as
religiously devout as the English statisticians. He was in
a runaway horse-drawn coach on a bridge and was saved

from drowning by the traces (the straps between the
horses and the carriage) breaking at the last possible mo-
ment. He took this as a warning to abandon his mathe-
matical work in favor of religious writings and later
formulated “Pascal’s wager”: that the value of a game is
the value of the prize times the probability of winning it;
therefore, even if the probability is low that God exists,
we should gamble on the affirmative because the value
of the prize is infinite, whereas the value of not believing
is only finite worldly pleasure.

Source: Tankard (1984).

T I P F O R S U C C E S S
To change a proportion into a
percentage, multiply by 100. So,
a proportion of .13 is equivalent to

. To change a
percentage into a proportion, di-
vide by 100. So, 3% is a propor-
tion of .3>100 = .03

.13 * 100 = 13%

Some Key Ingredients for Inferential Statistics 91

. It is also common to see probability written as being less than some
number, using the “less than” sign. For example, means “the probability is
less than .05.”

Probability Rules
As already noted, our discussion only scratches the surface of probability. One of the
topics we have not considered is the rules for figuring probabilities for multiple out-
comes: for example, what is the chance of flipping a coin twice and both times get-
ting heads? These are called probability rules, and they are important in the
mathematical foundation of many aspects of statistics. However, you don’t need to
know these probability rules to understand what we cover in this book. Also, the
rules are rarely used directly in analyzing results of psychology research. Neverthe-
less, you occasionally see such procedures referred to in research articles. Thus, the
most widely mentioned probability rules are described in the Advanced Topics sec-
tion toward the end of this chapter.

Probability, Z Scores, and the Normal Distribution
So far, we mainly have discussed probabilities of specific events that might or might
not happen. We also can talk about a range of events that might or might not happen.
The throw of a die coming out 3 or lower is an example (it includes the range 1, 2,
and 3). Another example is the probability of selecting someone on a city street who
is between the ages of 30 and 40.

If you think of probability in terms of the proportion of scores, probability fits in
well with frequency distributions (see Chapter 1). In the frequency distribution
shown in Figure 3–17, 3 of the total of 50 people scored 9 or 10. If you were select-
ing people from this group of 50 at random, there would be 3 chances (possible suc-
cessful outcomes) out of 50 (all possible outcomes) of selecting one that was 9 or 10.
Thus, .

You can also think of the normal distribution as a probability distribution. With
a normal curve, the percentage of scores between any two Z scores is known. The
percentage of scores between any two Z scores is the same as the probability of se-
lecting a score between those two Z scores. As you saw earlier in this chapter, ap-
proximately 34% of scores in a normal curve are between the mean and one standard
deviation from the mean. You should therefore not be surprised to learn that the
probability of a score being between the mean and a Z score of is about .34 (that
is, ).p = .34

+ 1

p = 3>50 = .06

p 6 .05
p = 50%

9
8
7
6
5
4
3
2
1

1 2 3 4 5 6 7 8 9 100

Figure 3–17 Frequency distribution (shown as a histogram) of 50 people, in which
of randomly selecting a person with a score of 9 or 10.p = .06 (3>50)

92 Chapter 3

In a previous IQ example in the normal curve section of this chapter, we fig-
ured that 95% of the scores in a normal curve are between a Z score of and
a Z score of (see Figure 3–14). Thus, if you were to select a score at ran-
dom from a distribution that follows a normal curve, the probability of selecting a
score between Z scores of and is .95 (that is, a 95% chance). This is
a very high probability. Also, the probability of selecting a score from such a dis-
tribution that is either greater than a Z score of or less than a Z score of

is .05 (that is, a 5% chance). This is a very low probability. It helps to think
about this visually. If you look back to Figure 3–14 on page 82, the .05 probabil-
ity of selecting a score that is either greater than a Z score of or less than a
Z score of is represented by the tail areas in the figure. The probability of a
score being in the tail of a normal curve is a topic you will learn more about in the
next chapter.

Probability, Samples, and Populations
Probability is also relevant to samples and populations. You will learn more about
this topic in Chapters 4 and 5, but we will use an example to give you a sense of the
role of probability in samples and populations. Imagine you are told that a sample
of one person has a score of 4 on a certain measure. However, you do not know
whether this person is from a population of women or of men. You are told that a
population of women has scores on this measure that are normally distributed with
a mean of 10 and a standard deviation of 3. How likely do you think it is that your
sample of 1 person comes from this population of women? From your knowledge
of the normal curve (see Figure 3–7), you know there are very few scores as low as
4 in a normal distribution that has a mean of 10 and a standard deviation of 3. So
there is a very low likelihood that this person comes from the population of women.
Now, what if the sample person had a score of 9? In this case, there is a much
greater likelihood that this person comes from the population of women because
there are many scores of 9 in that population. This kind of reasoning provides an in-
troduction to the process of hypothesis testing that is the focus of the remainder of
the book.

-1.96
+1.96
-1.96
+1.96

-1.96+1.96

-1.96
+1.96
How are you doing?

1. The probability of an event is defined as the expected relative frequency of
a particular outcome. Explain what is meant by (a) relative frequency and
(b) outcome.

2. List and explain two interpretations of probability.
3. Suppose you have 400 coins in a jar and 40 of them are more than 9 years

old. You then mix up the coins and pull one out. (a) What is the probability of
getting one that is more than 9 years old? (b) What is the number of possible
successful outcomes? (c) What is the number of all possible outcomes?

4. Suppose people’s scores on a particular personality test are normally distrib-
uted with a mean of 50 and a standard deviation of 10. If you were to pick a
person completely at random, what is the probability you would pick some-
one with a score on this test higher than 60?

5. What is meant by ?p 6 .01

Some Key Ingredients for Inferential Statistics 93

Controversies: Is the Normal Curve Really So
Normal? and Using Nonrandom Samples
Basic though they are, there is considerable controversy about the topics we have in-
troduced in this chapter. In this section we consider a major controversy about the
normal curve and nonrandom samples.

Is the Normal Curve Really So Normal?
We’ve said that real distributions in the world often closely approximate the normal
curve. Just how often real distributions closely follow a normal curve turns out to be
very important, not just because normal curves make Z scores more useful. As you
will learn in later chapters, the main statistical methods psychologists use assume
that the samples studied come from populations that follow a normal curve. Re-
searchers almost never know the true shape of the population distribution; so if they
want to use the usual methods, they have to just assume it is normal, making this as-
sumption because most populations are normal. Yet there is a long-standing debate
in psychology about just how often populations really are normally distributed. The
predominant view has been that, given how psychology measures are developed, a
bell-shaped distribution “is almost guaranteed” (Walberg et al., 1984, p. 107). Or, as
Hopkins and Glass (1978) put it, measurements in all disciplines are such good ap-
proximations to the curve that one might think “God loves the normal curve!”

On the other hand, there has been a persistent line of criticism about whether na-
ture really packages itself so neatly. Micceri (1989) showed that many measures
commonly used in psychology are not normally distributed “in nature.” His study in-
cluded achievement and ability tests (such as the SAT and the GRE) and personality
tests (such as the Minnesota Multiphasic Personality Inventory, MMPI). Micceri ex-
amined the distributions of scores of 440 psychological and educational measures
that had been used on very large samples. All of the measures he examined had been

Answers

1.(a) Relative frequency is the number of times something happens in relation to
the number of times it could have happened. (b) An outcome is the result of
an experiment—what happens in a situation where what will happen is not
known in advance.

2.(a) The long-run relative frequency interpretation of probability is that proba-
bility is the proportion of times we expect something to happen (relative to
how often it could happen) if the situation were repeated a very large number
of times. (b) The subjective interpretation of probability is that probability is
our sense of confidence that something will happen rated on a 0% to 100%
scale.

3.(a)The probability of getting one that is more than 9 years old is
(b) The number of possible successful outcomes is 40. (c) The number of all
possible outcomes is 400.

4.The probability you would pick someone with a score on this test higher than
60 is (since 16% of the scores are more than one standard deviation
above the mean).

5.The probability is less than .01.

p=.16

40>400 =.10.

94 Chapter 3

studied in samples of over 190 individuals, and the majority had samples of over
1,000 (14.3% even had samples of 5,000 to 10,293). Yet large samples were of no
help. No measure he studied had a distribution that passed all checks for normality
(mostly, Micceri looked for skewness, kurtosis, and “lumpiness”). Few measures
had distributions that even came reasonably close to looking like the normal curve.
Nor were these variations predictable: “The distributions studied here exhibited al-
most every conceivable type of contamination” (p. 162), although some were more
common with certain types of tests. Micceri discusses many obvious reasons for this
nonnormality, such as ceiling or floor effects (see Chapter 1).

How much has it mattered that the distributions for these measures were so far
from normal? According to Micceri, the answer is just not known. And until more is
known, the general opinion among psychologists will no doubt remain supportive of
traditional statistical methods, with the underlying mathematics based on the as-
sumption of normal population distributions.

What is the reason for this nonchalance in the face of findings such as Micceri’s?
It turns out that under most conditions in which the standard methods are used, they
give results that are reasonably accurate even when the formal requirement of a nor-
mal population distribution is not met (e.g., Sawilowsky & Blair, 1992). In this book,
we generally adopt this majority position favoring the use of the standard methods in
all but the most extreme cases. But you should be aware that a vocal minority of psy-
chologists disagrees. Some of the alternative statistical techniques they favor (ones
that do not rely on assuming a normal distribution in the population) are presented in
Chapter 14. These techniques include the use of nonparametric statistics that do not
have assumptions about the shape of the population distribution.

Francis Galton (1889), one of the major pioneers of statistical methods (see
Chapter 11, Box 11–1), said of the normal curve, “I know of scarcely anything so
apt to impress the imagination. . . . [It] would have been personified by the Greeks and
deified, if they had known of it. It reigns with serenity and in complete self-effacement
amidst the wild confusion” (p. 66). Ironically, it may be true that in psychology, at
least, it truly reigns in pure and austere isolation, with no even close-to-perfect real-
life imitators.

Using Nonrandom Samples
Most of the procedures you learn in the rest of this book are based on mathematics that
assume the sample studied is a random sample of the population. As we pointed out,
however, in most psychology research the samples are nonrandom, including whatev-
er individuals are available to participate in the experiment. Most studies are done
with college students, volunteers, convenient laboratory animals, and the like.

Some psychologists are concerned about this problem and have suggested that
researchers need to use different statistical approaches that make generalizations
only to the kinds of people that are actually being used in the study.3 For example,
these psychologists would argue that, if your sample has a particular nonnormal dis-
tribution, you should assume that you can generalize only to a population with the
same particular nonnormal distribution. We will have more to say about their sug-
gested solutions in Chapter 14.

Sociologists, as compared to psychologists, are much more concerned about the
representativeness of the groups they study. Studies reported in sociology journals
(or in sociologically oriented social psychology journals) are much more likely to
use formal methods of random selection and large samples, or at least to address the
issue in their articles.

Some Key Ingredients for Inferential Statistics 95

Why are psychologists more comfortable with using nonrandom samples? The
main reason is that psychologists are mainly interested in the relationships among
variables. If in one population the effect of experimentally changing X leads to a
change in Y, this relationship should probably hold in other populations. This rela-
tionship should hold even if the actual levels of Y differ from population to popula-
tion. Suppose that a researcher conducts an experiment testing the relation of
number of exposures to a list of words to number of words remembered. Suppose
further that this study is done with undergraduates taking introductory psychology
and that the result is that the greater the number of exposures is, the greater is the
number of words remembered. The actual number of words remembered from the
list might well be different for people other than introductory psychology students.
For example, chess masters (who probably have highly developed memories) may
recall more words; people who have just been upset may recall fewer words. How-
ever, even in these groups, we would expect that the more times someone is exposed
to the list, the more words will be remembered. That is, the relation of number of
exposures to number of words recalled will probably be about the same in each
population.

In sociology, the representativeness of samples is much more important. This is
because sociologists are more concerned with the actual mean and variance of a vari-
able in a particular society. Thus, a sociologist might be interested in the average at-
titude towards older people in the population of a particular country. For this
purpose, how sampling is done is extremely important.

Z Scores, Normal Curves, Samples and Populations,
and Probabilities in Research Articles
You need to understand the topics we covered in this chapter to learn what comes next.
However, the topics of this chapter are rarely mentioned directly in research articles
(except in articles about methods or statistics).Although Z scores are extremely impor-
tant as steps in advanced statistical procedures, they are rarely reported directly in
research articles. Sometimes you will see the normal curve mentioned, usually when a
researcher is describing the pattern of scores on a particular variable. (We say more
about this and give some examples from published articles in Chapter 14, where we
consider situations in which the scores do not follow a normal curve.)

Research articles will sometimes briefly mention the method of selecting the
sample from the population. For example, Viswanath and colleagues (2006) used
data from the U.S. National Cancer Institute (NCI) Health Information National
Trends Survey (HINTS) to examine differences in knowledge about cancer across
individuals from varying socioeconomic and racial/ethnic groups. They described
the method of their study as follows:

The data from this study come from the NCI HINTS, based on a random-digit-dial
(RDD) sample of all working telephones in the United States. One adult was selected
at random within each household using the most recent birthday method in the case of
more than three adults in a given household. . . . Vigorous efforts were made to increase
response rates through advanced letters and $2 incentives to households. (p. 4)

Whenever possible, researchers report the proportion of individuals approached for
the study who actually participated in the study. This is called the response rate.
Viswanath and colleagues (2006) noted that “The final sample size was 6,369, yield-
ing a response rate of 55%” (p. 4).

96 Chapter 3

Researchers sometimes also check whether their sample is similar to the popu-
lation as a whole, based on any information they may have about the overall popula-
tion. For example, Schuster and colleagues (2001) conducted a national survey of
stress reactions of U.S. adults after the September 11, 2001, terrorist attacks. In this
study, the researchers compared their sample to 2001 census records and reported
that the “sample slightly overrepresented women, non-Hispanic whites, and persons
with higher levels of education and income” (p. 1507). Schuster and colleagues went
on to note that overrepresentation of these groups “is typical of samples selected by
means of random-digit dialing” (pp. 1507–1508).

However, even survey studies typically are not able to use such rigorous meth-
ods and have to rely on more haphazard methods of getting their samples. For exam-
ple, in a study of relationship distress and partner abuse (Heyman et al., 2001), the
researchers describe their method of gathering research participants to interview as
follows: “Seventy-four couples of varying levels of relationship adjustment were re-
cruited through community newspaper advertisements” (p. 336). In a study of this
kind, one cannot very easily recruit a random sample of abusers since there is no list
of all abusers to recruit from! This could be done with a very large national random
sample of couples, who would then include a random sample of abusers. Indeed, the
authors of this study are very aware of the issues. At the end of the article, when dis-
cussing “cautions necessary when interpreting our results,” they note that before
their conclusions can be taken as definitive “our study must be replicated with a rep-
resentative sample” (p. 341).

Finally, probability is rarely discussed directly in research articles, except in rela-
tion to statistical significance, a topic we discuss in the next chapter. In almost any ar-
ticle you look at, the results section will be strewn with descriptions of various
methods having to do with statistical significance, followed by something like
“ ” or “ .” The p refers to probability, but the probability of what? This
is the main topic of our discussion of statistical significance in the next chapter.

Advanced Topic: Probability Rules
and Conditional Probabilities
This advanced topic section provides additional information on probability, focusing
specifically on probability rules and conditional probabilities. Probability rules are pro-
cedures for figuring probabilities in more complex situations than we have considered
so far. This section considers the two most widely used such rules and also explains the
concept of conditional probabilities that is used in advanced discussions of probability.

Addition Rule
The addition rule (also called the or rule) is used when there are two or more
mutually exclusive outcomes. “Mutually exclusive” means that, if one outcome hap-
pens, the others can’t happen. For example, heads or tails on a single coin flip are
mutually exclusive because the result has to be one or the other, but can’t be both.
With mutually exclusive outcomes, the total probability of getting either outcome is
the sum of the individual probabilities. Thus, on a single coin flip, the total chance of
getting either heads (which is .5) or tails (also .5) is 1.0 (.5 plus .5). Similarly, on a
single throw of a die, the chance of getting either a 3 (1�6) or a 5 (1�6) is

. If you are picking a student at random from your university in
which 30% are seniors and 25% are juniors, the chance of picking someone who is
either a senior or a junior is 55%.

1>3 (1>6 + 1>6)

p 6 .01p 6 .05

Some Key Ingredients for Inferential Statistics 97

Even though we have not used the term addition rule, we have already used
this rule in many of the examples we considered in this chapter. For example, we
used this rule when we figured that the chance of getting a 3 or lower on the throw
of a die is .5.

Multiplication Rule
The multiplication rule (also called the and rule), however, is completely new. You
use the multiplication rule to figure the probability of getting both of two (or more)
independent outcomes. Independent outcomes are those for which getting one has no
effect on getting the other. For example, getting a head or tail on one flip of a coin is
an independent outcome from getting a head or tail on a second flip of a coin. The
probability of getting both of two independent outcomes is the product of (the result
of multiplying) the individual probabilities. For example, on a single coin flip, the
chance of getting a head is .5. On a second coin flip, the chance of getting a head (re-
gardless of what you got on the first flip) is also .5. Thus, the probability of getting
heads on both coin flips is .25 (.5 multiplied by .5). On two throws of a die, the
chance of getting a 5 on both throws is 1�36—the probability of getting a 5 on the
first throw (1�6) multiplied by the probability of getting a 5 on the second throw
(1�6). Similarly, on a multiple choice exam with four possible answers to each item,
the chance of getting both of two questions correct just by guessing is 1�16—that is,
the chance of getting one question correct just by guessing (1�4) multiplied by the
chance of getting the other correct just by guessing (1�4). To take one more example,
suppose you have a 20% chance of getting accepted into one graduate school and a
30% chance of getting accepted into another graduate school. Your chance of getting
accepted at both graduate schools is just 6% (that is, ).

Conditional Probabilities
There are several other probability rules, some of which are combinations of the ad-
dition and multiplication rules. Most of these other rules have to do with what are
called conditional probabilities. A conditional probability is the probability of
one outcome, assuming some other outcome will happen. That is, the probability of
the one outcome depends on—is conditional on—the probability of the other out-
come. Thus, suppose that college A has 50% women and college B has 60% women.
If you select a person at random, what is the chance of getting a woman? If you
know the person is from college A, the probability is 50%. That is, the probability of
getting a woman, conditional upon her coming from college A, is 50%.

20% * 30% = 6%

1. A Z score is the number of standard deviations that a raw score is above or
below the mean.

2. The scores on many variables in psychology research approximately follow a
bell-shaped, symmetrical, unimodal distribution called the normal curve. Be-
cause the shape of this curve follows an exact mathematical formula, there is a
specific percentage of scores between any two points on a normal curve.

3. A useful working rule for normal curves is that 50% of the scores are above the
mean, 34% are between the mean and 1 standard deviation above the mean, and
14% are between 1 and 2 standard deviations above the mean.

Summary

98 Chapter 3

4. A normal curve table gives the percentage of scores between the mean and any
particular Z score, as well as the percentage of scores in the tail for any Z score.
Using this table, and knowing that the curve is symmetrical and that 50% of the
scores are above the mean, you can figure the percentage of scores above or
below any particular Z score. You can also use the table to figure the Z score for
the point where a particular percentage of scores begins or ends.

5. A sample is an individual or group that is studied—usually as representative of
some larger group or population that cannot be studied in its entirety. Ideally, the
sample is selected from a population using a strictly random procedure.
The mean (M), variance , standard deviation (SD), and so forth of a sam-
ple are called sample statistics. When of a population, the sample statistics are
called population parameters and are symbolized by Greek letters—� for mean,

for variance, and � for standard deviation.
6. Most psychology researchers consider the probability of an event to be its ex-

pected relative frequency. However, some think of probability as the subjective
degree of belief that the event will happen. Probability is figured as the propor-
tion of successful outcomes to total possible outcomes. It is symbolized by p
and has a range from 0 (event is impossible) to 1 (event is certain). The normal
curve provides a way to know the probabilities of scores being within particular
ranges of values.

7. There are controversies about many of the topics in this chapter. One is about
whether normal distributions are truly typical of the populations of scores for
the variables we study in psychology. In another controversy, some researchers
have questioned the use of standard statistical methods in the typical psychology
research situation that does not use strict random sampling.

8. Research articles rarely discuss Z scores, normal curves (except briefly when
a variable being studied seems not to follow a normal curve), or probability
(except in relation to statistical significance). Procedures of sampling, particu-
larly when the study is a survey, are sometimes described, and the representa-
tiveness of a sample may also be discussed.

9. ADVANCED TOPIC: In situations where there are two or more mutually exclu-
sive outcomes, probabilities are figured following an addition rule, in which the
total probability is the sum of the individual probabilities. A multiplication rule
(in which probabilities are multiplied together) is followed to figure the proba-
bility of getting both of two (or more) independent outcomes. A conditional
probability is the probability of one outcome, assuming some other particular
outcome will happen.

�2

(SD2)

Z score (p. 68)
raw score (p. 69)
normal distribution (p. 73)
normal curve (p. 73)
normal curve table (p.76)
population (p. 83)
sample (p. 83)

random selection (p. 85)
population parameters (p. 87)
� (population mean) (p. 87)
�2 (population variance) (p. 87)
� (population standard

deviation) (p. 87)
sample statistics (p. 87)

probability (p) (p. 89)
outcome (p. 89)
expected relative frequency (p. 89)
long-run relative-frequency interpre-

tation of

probability (p. 89)

subjective interpretation of

probability (p. 89)

Key Terms

Some Key Ingredients for Inferential Statistics 99

Changing a Raw Score to a Z Score
A distribution has a mean of 80 and a standard deviation of 20. Find the Z score for
a raw score of 65.

Answer
You can change a raw score to a Z score using the formula or the steps.

Using the formula: .
Using the steps:

❶ Figure the deviation score: subtract the mean from the raw score.
.

❷ Figure the Z score: divide the deviation score by the standard deviation.
.

Changing a Z Score to a Raw Score
A distribution has a mean of 200 and a standard deviation of 50. A person has a Z
score of 1.26. What is the person’s raw score?

Answer
You can change a Z score to a raw score using the formula or the steps.

Using the formula: .
Using the steps:

❶ Figure the deviation score: multiply the Z score by the standard deviation.
.

❷ Figure the raw score: add the mean to the deviation score.

Outline for Writing Essays Involving Z Scores
1. If required by the question, explain the mean, variance, and standard deviation

(using the points in the essay outlined in Chapter 2).
2. Describe the basic idea of a Z score as a way of describing where a particular

score fits into an overall group of scores. Specifically, a Z score shows the num-
ber of standard deviations a score is above or below the mean.

3. Explain the steps for figuring a Z score from a raw (ordinary) score.
4. Mention that changing raw scores to Z scores puts scores that are for different

variables onto the same scale, which makes it easier to make comparisons be-
tween scores on the variables.

Figuring the Percentage Above or Below a Particular Raw
Score or Z Score
Suppose a test of sensitivity to violence is known to have a mean of 20, a standard
deviation of 3, and a normal curve shape. What percentage of people have scores
above 24?

Answer
❶ If you are beginning with a raw score, first change it to a Z score. Using the

usual formula, .Z = (X – M)>SD, Z = (24 – 20)>3 = 1.33

63 + 200 = 263.
1.26 * 50 = 63

X = (Z)(SD) + M = (1.26)(50) + 200 = 63 + 200 = 263

-15>20 = .75

65 – 80 = -15

Z = (X – M)>SD = (65 – 80)>20 = -15>20 = – .75

Example Worked-Out Problems

100 Chapter 3

❷ Draw a picture of the normal curve, decide where the Z score falls on it, and
shade in the area for which you are finding the percentage. This is shown in
Figure 3–18.

❸ Make a rough estimate of the shaded area’s percentage based on the
50%–34%–14% percentages. If the shaded area started at a Z score of 1, it
would include 16%. If it started at a Z score of 2, it would include only 2%. So
with a Z score of 1.33, it has to be somewhere between 16% and 2%.

❹ Find the exact percentage using the normal curve table, adding 50% if nec-
essary. In Table A–1 (in the Appendix), 1.33 in the “Z ” column goes with
9.18% in the “% in Tail” column. This is the answer to our problem: 9.18% of
people have a higher score than 24 on the sensitivity to violence measure.
(There is no need to add 50% to the percentage.)

❺ Check that your exact percentage is within the range of your rough esti-
mate from Step ❸. Our result, 9.18%, is within the 16% to 2% range estimated.

Note: If the problem involves Z scores that are all 0, 1, or , you can
work the problem using the figures and without using the normal
curve table (although you should still draw a figure and shade in the appropriate
area).

Figuring Z Scores and Raw Scores From Percentages
Consider the same situation: A test of sensitivity to violence is known to have a mean
of 20, a standard deviation of 3, and a normal curve shape. What is the minimum
score a person needs to be in the top 75%?

Answer
❶ Draw a picture of the normal curve, and shade in the approximate area for

your percentage using the 50%–34%–14% percentages. The shading has to
begin between the mean and 1 SD below the mean. (There are 50% above the
mean and 84% above 1 SD below the mean). This is shown in Figure 3–19.

❷ Make a rough estimate of the Z score where the shaded area stops. The Z
score has to be between 0 and .

❸ Find the exact Z score using the normal curve table (subtracting 50% from
your percentage if necessary before looking up the Z score). Since 50% of
people have IQs above the mean, for the top 75% you need to include the 25%
below the mean (that is, ). Looking in the “% Mean to Z ”75% – 50% = 25%

-1

50%–34%–14%

2 (or -1 or -2)

20
0

50%
9.18%

+1.33

Figure 3–18 Distribution of sensitivity to violence scores showing the percentage of
scores above a score of 24 (shaded area).

Some Key Ingredients for Inferential Statistics 101

column of the normal curve table, the closest figure to 25% is 24.86, which goes
with a Z of .67. Since we are interested in below the mean, we want .

❹ Check that your exact Z score is within the range of your rough estimate
from Step ❷. is between 0 and .

❺ If you want to find a raw score, change it from the Z score. Using the formula
. That is, to

be in the top 75%, a person needs to have a score on this test of at least 18.

Note: If the problem instructs you not to use a normal curve table, you should be able
to work the problem using the figures (although you should still
draw a figure and shade in the appropriate area).

Outline for Writing Essays on the Logic and Computations
for Figuring a Percentage from a Z Score and Vice Versa

1. Note that the normal curve is a mathematical (or theoretical) distribution, describe
its shape (be sure to include a diagram of the normal curve), and mention that
many variables in nature and in research approximately follow a normal curve.

2. If required by the question, explain the mean and standard deviation (using the
points in the essay outline in Chapter 2).

3. Describe the link between the normal curve and the percentage of scores be-
tween the mean and any Z score. Be sure to include a description of the normal
curve table and show how it is used.

4. Briefly describe the steps required to figure a percentage from a Z score or vice
versa (as required by the question). Be sure to draw a diagram of the normal
curve with appropriate numbers and shaded areas marked on it from the relevant
question (e.g., the mean, one and two standard deviations above/below the
mean, shaded area for which percentage or Z score is to be determined).

Finding a Probability
A candy dish has four kinds of fruit-flavored candy: 20 apple, 20 strawberry, 5 cherry,
and 5 grape. If you close your eyes and pick one piece of candy at random, what is
the probability it will be either cherry or grape?

Answer
❶ Determine the number of possible successful outcomes. There are 10 possible

successful outcomes—5 cherry and 5 grape.

50%–34%–14%

X = (Z)(SD) + M, X = (- .67)(3) + 20 = -2.01 + 20 =

17.99

-1

– .67

–2

–1

25% 50%

17.99

–.67

20
0
23
+1
26
+2

75%

Figure 3–19 Finding the sensitivity to violence raw score for where the top 75% of
scores start.

102 Chapter 3

❷ Determine the number of all possible outcomes. There are 50 possible out-
comes overall: .

❸ Divide the number of possible successful outcomes (Step ❶) by the number
of all possible outcomes (Step ❷). . Thus, the probability of picking
either a cherry or grape candy is .2.

10>50 = .2

20 + 20 + 5 + 5 = 50

Practice Problems

All data are fictional unless an actual citation is given.

Set I (for Answers to Set I Practice Problems, see p. 675)
1. On a measure of anxiety, the mean is 79 and the standard deviation is 12. What

are the Z scores for each of the following raw scores? (a) 91, (b) 68, and (c) 103.
2. On an intelligence test, the mean number of raw items correct is 231 and the

standard deviation is 41. What are the raw (actual) scores on the test for people
with IQs of (a) 107, (b) 83, and (c) 100? To do this problem, first figure the Z
score for the particular IQ score; then use that Z score to find the raw score. Note
that IQ scores have a mean of 100 and a standard deviation of 16.

3. Six months after a divorce, the former wife and husband each take a test that
measures divorce adjustment. The wife’s score is 63, and the husband’s score is
59. Overall, the mean score for divorced women on this test is 60 ; the
mean score for divorced men is 55 . Which of the two has adjusted
better to the divorce in relation to other divorced people of the same gender? Ex-
plain your answer to a person who has never had a course in statistics.

4. Suppose the people living in a city have a mean score of 40 and a standard devi-
ation of 5 on a measure of concern about the environment. Assume that these
concern scores are normally distributed. Using the figures, ap-
proximately what percentage of people have a score (a) above 40, (b) above 45,
(c) above 30, (d) above 35, (e) below 40, (f) below 45, (g) below 30, and (h)
below 35?

5. Using the information in problem 4 and the figures, what is
the minimum score a person has to have to be in the top (a) 2%, (b) 16%,
(c) 50%, (d) 84%, and (e) 98%?

6. A psychologist has been studying eye fatigue using a particular measure, which
she administers to students after they have worked for 1 hour writing on a com-
puter. On this measure, she has found that the distribution follows a normal
curve. Using a normal curve table, what percentage of students have Z scores
(a) below 1.5, (b) above 1.5, (c) below , (d) above , (e) above 2.10,
(f) below 2.10, (g) above .45, (h) below , and (i) above 1.68?

7. In the previous problem, the test of eye fatigue has a mean of 15 and a standard
deviation of 5. Using a normal curve table, what percentage of students have
scores (a) above 16, (b) above 17, (c) above 18, (d) below 18, (e) below 14?

-1.78
-1.5-1.5

50%–34%–14%
50%–34%–14%

(SD = 4)
(SD = 6)

Some Key Ingredients for Inferential Statistics 103

8. In the eye fatigue example of problems 6 and 7, using a normal curve table,
what is the lowest score on the eye fatigue measure a person has to have to be in
(a) the top 40%, (b) the top 30%, (c) the top 20%?

9. Using a normal curve table, give the percentage of scores between the mean and
a Z score of (a) .58, (b) .59, (c) 1.46, (d) 1.56, (e) .

10. Consider a test of coordination that has a normal distribution, a mean of 50, and
a standard deviation of 10. (a) How high a score would a person need to be in
the top 5%? (b) Explain your answer to someone who has never had a course in
statistics.

11. Altman et al. (1997) conducted a telephone survey of the attitudes of the U.S.
adult public toward tobacco farmers. In the method section of their article, they
explained that their respondents were “randomly selected from a nationwide list
of telephone numbers” (p. 117). Explain to a person who has never had a course
in statistics or research methods what this means and why it is important.

12. The following numbers of individuals in a company received special assistance
from the personnel department last year:

– .58

Drug/alcohol 10

Family crisis counseling 20

Other 20

Total 50

If you were to select someone at random from the records for last year, what is
the probability that the person would be in each of the following categories:
(a) drug/alcohol, (b) family, (c) drug/alcohol or family, (d) any category except
“Other,” or (e) any of the three categories? (f) Explain your answers to someone
who has never had a course in statistics.

Set II
13. On a measure of artistic ability, the mean for college students in New Zealand is

150 and the standard deviation is 25. Give the Z scores for New Zealand college
students who score (a) 100, (b) 120, (c) 140, and (d) 160. Give the raw scores
for persons whose Z scores on this test are (e) , (f) , (g) , and
(h) .

14. On a standard measure of hearing ability, the mean is 300 and the standard devi-
ation is 20. Give the Z scores for persons who score (a) 340, (b) 310, and
(c) 260. Give the raw scores for persons whose Z scores on this test are (d) 2.4,
(e) 1.5, (f) 0, and (g) .

15. A person scores 81 on a test of verbal ability and 6.4 on a test of quantitative
ability. For the verbal ability test, the mean for people in general is 50 and the
standard deviation is 20. For the quantitative ability test, the mean for people in
general is 0 and the standard deviation is 5. Which is this person’s stronger abil-
ity: verbal or quantitative? Explain your answer to a person who has never had a
course in statistics.

16. The amount of time it takes to recover physiologically from a certain kind of
sudden noise is found to be normally distributed with a mean of 80 seconds and
a standard deviation of 10 seconds. Using the figures, approx-
imately what percentage of scores (on time to recover) will be (a) above 100,
(b) below 100, (c) above 90, (d) below 90, (e) above 80, (f) below 80, (g) above
70, (h) below 70, (i) above 60, and (j) below 60?

50%–34%–14%

-4.5

+1.38
– .2- .8-1

104 Chapter 3

17. Using the information in problem 16 and the figures, what is
the longest time to recover that a person can take and still be in the bottom
(a) 2%, (b) 16%, (c) 50%, (d) 84%, and (e) 98%?

18. Suppose that the scores of architects on a particular creativity test are normally
distributed. Using a normal curve table, what percentage of architects have Z
scores (a) above .10, (b) below .10, (c) above .20, (d) below .20, (e) above 1.10,
(f) below 1.10, (g) above , and (h) below ?

19. In the example in problem 18, using a normal curve table, what is the minimum
Z score an architect can have on the creativity test to be in the (a) top 50%,
(b) top 40%, (c) top 60%, (d) top 30%, and (e) top 20%?

20. In the example in problem 18, assume that the mean is 300 and the standard de-
viation is 25. Using a normal curve table, what scores would be the top and bot-
tom scores to find (a) the middle 50% of architects, (b) the middle 90% of
architects, and (c) the middle 99% of architects?

21. Suppose that you are designing an instrument panel for a large industrial machine.
The machine requires the person using it to reach 2 feet from a particular position.
The reach from this position for adult women is known to have a mean of 2.8 feet
with a standard deviation of .5. The reach for adult men is known to have a mean
of 3.1 feet with a standard deviation of .6. Both women’s and men’s reach from
this position is normally distributed. If this design is implemented, (a) what per-
centage of women will not be able to work on this instrument panel? (b) What per-
centage of men will not be able to work on this instrument panel? (c) Explain your
answers to a person who has never had a course in statistics.

22. Suppose you want to conduct a survey of the attitude of psychology graduate stu-
dents studying clinical psychology toward psychoanalytic methods of psychother-
apy. One approach would be to contact every psychology graduate student you
know and ask them to fill out a questionnaire about it. (a) What kind of sampling
method is this? (b) What is a major limitation of this kind of approach?

23. A large study of how people make future plans and the relation of this to their
life satisfaction (Prenda & Lachman, 2001) recruited participants “through
random-digit dialing procedures.” These are procedures in which phone num-
bers to call potential participants are randomly generated by a computer. Ex-
plain to a person who has never had a course in statistics (a) why this method of
sampling might be used and (b) why it may be a problem if not everyone called
agreed to be interviewed.

24. Suppose that you were going to conduct a survey of visitors to your campus.
You want the survey to be as representative as possible. (a) How would you se-
lect the people to survey? (b) Why would that be your best method?

25. You are conducting a survey at a college with 800 students, 50 faculty members,
and 150 administrators. Each of these 1,000 individuals has a single listing in
the campus phone directory. Suppose you were to cut up the directory and pull
out one listing at random to contact. What is the probability it would be (a) a stu-
dent, (b) a faculty member, (c) an administrator, (d) a faculty member or admin-
istrator, and (e) anyone except an administrator? (f) Explain your answers to
someone who has never had a course in statistics.

26. You apply to 20 graduate programs, 10 of which are in clinical psychology, 5 of
which are in counseling psychology, and 5 of which are in social work. You get
a message from home that you have a letter from one of the programs you ap-
plied to, but nothing is said about which one. Give the probabilities it is from (a)
a clinical psychology program, (b) a counseling psychology program, (c) from
any program other than social work. (d) Explain your answers to someone who
has never had a course in statistics.

– .10- .10

Some Key Ingredients for Inferential Statistics 105

The U in the following steps indicates a mouse click. (We used SPSS version 15.0
to carry out these analyses. The steps and output may be slightly different for other
versions of SPSS.)

Changing Raw Scores to Z Scores
It is easier to learn these steps using actual numbers, so we will use the number of
dreams example from Chapter 2.

❶ Enter the scores from your distribution in one column of the data window (the
scores are 7, 8, 8, 7, 3, 1, 6, 9, 3, 8). We will call this variable “dreams.”

❷ Find the mean and standard deviation of the scores. You learned how to do this
in the Chapter 2 Using SPSS section (see p. 62). The mean is 6 and the standard
deviation is 2.57.

❸ You are now going to create a new variable that shows the Z score for each raw
score. U Transform, U Compute Variable. You can call the new variable any
name that you want, but we will call it “zdreams.” So, write zdreams in the box
labeled Target Variable. In the box labeled Numeric Expression, write (dreams �
6)�2.57. As you can see, this formula creates a deviation score (by subtracting
the mean from the raw score) and divides the deviation score by the standard de-
viation. U OK. You will see that a new variable called zdreams has been added
to the data window. The scores for this zdreams variable are the Z scores for the
dreams variable.4 Your data window should now look like Figure 3–20.

Using SPSS

Figure 3–20 Using SPSS to change raw scores to Z scores for the number of dreams
example.

106 Chapter 3

Chapter Notes

1. Also, sometimes used are scores similar to Z scores, called T scores, in which
the mean is 50 and the standard deviation is 10. For example, some tests used by
clinical psychologists use a T score scale. Thus, a 65 on a scale of T scores
equals a Z score of 1.5.

2. The formula for the normal curve (when the mean is 0 and the standard devia-
tion is 1) is

where is the height of the curve at point x and � and e are the usual mathe-
matical constants (approximately 3.14 and 2.72, respectively). However, psy-
chology researchers almost never use this formula because it is built into the
statistics software that do calculations involving normal curves. When work
must be done by hand, any needed information about the normal curve is pro-
vided in tables in statistics books (for example, Table A–1 in the Appendix).

3. Frick (1998) argued that in most cases psychology researchers should not think
in terms of samples and populations at all. Rather, he argues, researchers should
think of themselves as studying processes. An experiment examines some
process in a group of individuals. Then the researcher evaluates the probability
that the pattern of results could have been caused by chance factors. For exam-
ple, the researcher examines whether a difference in means between an experi-
mental and a control group could have been caused by factors other than by the
experimental manipulation. Frick claims that this way of thinking is much
closer to the way researchers actually work, and argues that it has various
advantages in terms of the subtle logic of inferential statistical procedures.

4. You can also request the Z scores directly from SPSS. However, SPSS figures
the standard deviation based on the “dividing by formula” for the vari-
ance (see Chapters 2 and 6). Thus, the Z scores figured directly by SPSS will be
different from the Z scores as you learn to figure them. Here are the steps for fig-
uring Z scores directly from SPSS: ❶ Enter the scores from your distribution
in one column of the data window. ❷ U Analyze, U Descriptive statistics,
U Descriptives. ❸ U on the variable for which you want to find the Z scores,
and then U the arrow. ❹ U the box labeled Save standardized values as vari-
ables (this checks the box). ❺ U OK. A new variable is added to the data win-
dow. The values for this variable are the Z scores for your variable (based on the
dividing by formula). (You can ignore the output window, which by
default will show descriptive statistics for your variable.)

N – 1

f(x)

f(x) =
1

22�
e-x

2
> 2

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Statistics for Psychology

Chapter 2 Instructions

Mdn. =

SS =

SD =

CEO’s – M = SD =

Name:

Chapter 3 Instructions

Quantitative Ability Z =

CEO’s
–
M =
SD =