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excel for Statistics class

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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.

CHAPTER

Central Tendency and Variability

Chapter Outline

C Central Tendency 3

C Variability 4

C Controversy: The Tyranny
of the Mean 52

Central Tendency and Variability
in Research Articles 5

O Summary 5

C Key Terms 57

C Example Worked-Out Problems 57
C Practice Problems 5

C Using SPSS 62

C Chapter Notes 65

As 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.

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.

M =

1 2 3 4 5 6 7

8 9

Chapter 2

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

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

Central Tendency
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 (1 + 1 + 2 + 2 + 2 + 3) 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 (3 + 3 + 5) 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).

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 and Variability 35

M = 6 M = 6

1 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7 8 9

M = 3.60 M = 6

1 2 3 4 5 6 7 8 9 I 2 3 4 5 6 7

8 9

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

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, this
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:

M = EX (2-1) 4 The mean is the sum of the scores divided by the number
of scores.

M is a symbol for the mean. An alternative symbol, X (“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 X 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.)

E, 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 X 1 and X 2 ).

EX 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: E X is
7 + 8 + 8 + 7 + 3 + 1 +6+9+3+ 8, which is 60.

Eh vit fii.aiiir44
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.

M mean.

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

X scores in the distribution of the
variable X.

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 formul

Ex 6

M = = — = 6

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

X193
M– = =

6.43

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.

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.

N number of scores in a distribution.

0
6.43

Balance Point

0 1 2 3

Stress 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.)

Central Tendency and Variability 37

cr 7
1:1
▪ 6

5
4
3
2

O i
2.5 7.5 12.5 22.5 27.5 32.5 37.5 42.5 47.5

17.39

Balance Point

Number of Social Interactions

in a Week

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.)

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:

EX 1,635

M = —

N 94

= 17.39

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.

O Add up all the scores. That is, figure F.X.
• Divide this sum by the number of scores. That is, divide EX 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).

mode value with the greatest frequency
in a distribution.

Mode = 8

1 2 3 4 5 6 7 8

38 Chapter 2

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.

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).

Mode = 8

Mean = 8.30

2 3 4 5 6 7 8 9 10 11

Mode = 8

Mean = 5.10

2 3 4 5 6 7 8 9 10

Mode = 8

Mean = 7

2 3 4 5 6 7 8 9 10

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.

Central Tendency and Variability 39

1 3 3 6 7 7 8 8 8 9

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.

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.

O Line up all the scores from lowest to highest.
@ 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 25 11/2. Because there are no half scores, the
25th and 26th scores (the scores on either side of 251/2) are the middle scores.

• 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, (IX)/N = 5.52/5 = 1.1040. 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.

. _TIP FOR SUCCESS
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.

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

Men

Source: Data from Miller & Fishkin (1997).

0 Men %
■ Women %

Measures of central tendency
Mean Median

Men

64.3 1
Women 2.8 1

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. ],, 1-1
00 0

`’
I.
7 r-

00 os
1 7

.. ∎O r•.- 00 “J;

Number of Partners Desired in the Next 30 Years

Pe
rc

en
ta

ge
o

f M
en

a
nd

W
om

20 –

10 –

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.

Mean
Mode

Median

Central Tendency and Variability 41

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.

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.

Chapter 2

Table 2-2 Summary of Measures of Central Tendency

Measure Definition When Used

Mean

Sum of the scores divided by the
number of scores

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

research

Mode
Median

Value with the greatest frequency
in a distribution

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

• With nominal variables
• Rarely used in psychology research

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

outliers
• Rarely used in psychology research

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, psychologists
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.

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?

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Mean

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3.20
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2.50
Mean

3.20
Mean

Central Tendency and Variability 43

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

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.

Neighborhood with
Diverse
Types of Housing

Housing
Prices

Mean

Neighborhood with
Consistent
Type of Housing

Housing Mean
Prices

Chapter 2

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.

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:

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

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

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

0 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.

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.

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

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 29 — 29 = 0; 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 19 — 29 = —10. The 39-year-olds would have deviation
scores of 39 — 29 = 10. All the squared deviation scores, which are either —10
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.

0 Figure the variance.
(9 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 +3 and —3.)

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.

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.

SD 2 = 1(X M)2 (2-2) 4 The variance is the sum of the squared deviations of the scores from the mean, divided
by the number of scores.

46 Chapter 2

“1-
4

I
3

I
5

I
0

I
1

I
2

I
3

I
4

I
5

I
6

I
7

I
0

1
1

I
2
I
6

1
5

1
6

1
7

I
0
I
1

1
2

I
3
I
4
I
5
I
6
I
7

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

SD 2 is the symbol for the variance. This may seem surprising. SD is short for
standard deviation. The symbol SD 2 emphasizes that the variance is the standard devi-
ation squared. (Later, you will learn other symbols for the variance, S 2 and o-2—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
(1) 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 E(X – M) 2 :

SD2 variance.

SD standard deviation.

sum of squares (SS) sum of squared
deviations.

vim lor-o I 411s-M1
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.

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

SD 2 = SS

(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.

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.

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.

11111 IF • MP 1 ./T 4 ir -g -SIM

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.

MI II -fog Awl rota -1 -C411.
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!

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

SD = VSD 2 (2-4)

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

SD = E(X – M) 2

N
or

SS
SD = V ITT

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.

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 —3 9

1 6 —5 25

6 6 0 0

9 6 3 9

3 6 —3 9
8 6 2 4

2,:0 66

E (X —M) 2 SS 66
Variance = SD’ = — — —

N 10
= 6.60

Standard deviation = SD = \/SD 2 = 16.60 = 2.57

(2-5)

(2-6)

16
15

1 4

13
12
11
10
9
8
7
6
5
4
3
2

2.5 7.5 12.5 17.5 22.5 27.5 32.5 37.5 42.5 47.5

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 -2.39 5.71

33 17.39 15.61 243.67

3 17.39 -14.39 207.07

21 17.39 3.61 13.03

35 17.39 17.61 310.11

9 17.39 -8.39 70.39

30 17.39 12.61 159.01

8 17.39 -9.39 88.17

26 17.39 8.61 74.13

E: 0.00 12,406.44

SD 2
E (X – M) 2 12,406.44

Variance = SD` =
N

=
94

– 131.98

Standard deviation = VSD 2 = V131.98 = 11.49

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

INTERVAL FREQUENCY
0 – 4 12
5 – 9 16

10 – 14 16
15 – 19 16
20 – 24 10
25 – 29 11
30 – 34 4
35 – 39 3
40 – 44 3
45 – 49 3

4
1 SD
)• 1

1SD
M

Number of Social Interactions in a Week

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.)

1111111711,111TINTITIMITIIIIIPP

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 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.

Central Tendency and Variability 49

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:

— ((EX) 2/N)
SD2 = (2- 7)

EX 2 means that you square each score and then take the sum of the squared
scores. However, ( X ) 2 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,

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.

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 N — 1
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 I less than the number of
scores. In other words, for those purposes the variance is the sum of squared deviations
divided by N — 1 (that is, variance is SS/[N — 1]). (As you will learn in Chapter 7,
you use this dividing by N — 1 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/(N — 1). 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/[N — 1]) 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 N — 1 approach until it is needed, starting in Chapter 7.

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 (M = 4).

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?

Central Tendency and Variability 51

– N)/SS ‘S! le141)

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SJ3MSUV

BOX 2-1 The Sheer Joy (Yes, Joy) of Statistical Analysis
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 1 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

52 Chapter 2

. . . 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.

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

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.

BOX 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).

54 Chap ter 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.

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 -.01 .81 .65

Mobile phone 399 -.19 .88 .77

Landline telephone 404 -.77 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.

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 (SD = 2.57).” 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 -3 for very fem-
inine to +3 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

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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, M = (EX)/ N .

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, SD 2 = [ 1(X — M) 21/N. The sum of squared deviations,
E(X — M) 2 , is also symbolized as SS. Thus SD 2 = SS/N.

5. The standard deviation is the square root of the variance. In symbols,
SD = VSD 2 . 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.

central tendency (p. 34)
mean (M) (pp. 34, 35)

(sum of) (p. 35)
X (p. 35)
N (number of scores) (p. 36)
mode (p. 37)
median (p. 39)

outlier (p. 39)
variance (SD 2 ) (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)

SD 2 (p. 46)
SD (p. 46)
computational formula (p. 49)
definitional formula (p. 49)

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: M = (E X)/ N = 48/8 = 6
Using the steps:

0 Add up all the scores. 8 + 6 + 6 + 9 + 6 + 5 + 6 + 2 = 48.
@ Divide this sum by the number of scores. 48/8 = 6.

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

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 1

6 6 0 0

2 6 —4 16

= SS = 30

0 Variance = 30/8 = 3.75

Chapter 2

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

Answer

O Line up all the scores from lowest to highest. 1, 2, 2, 3, 4, 6, 7, 7, 9.
• 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.

• 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: M = 6.)

Answer

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

SS = (X — M)2 = (8 — 6) 2 + (6 — 6) 2 + (6 — 6) 2
+ (9 — 6) 2 + (6 — 6) 2 + (5 — 6) 2 + (6 — 6) 2 + (2 — 6) 2

= 22 + 02 + 02 + 32 + 02 + _12 + 02 + 42

= 4 + 0 + 0 + 9 + 0 + 1 + 0 + 16
= 30

SD 2 = SS/ N = 30/8 = 3.75.

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

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

variance (SD 2 ).

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. SD 2 = 3.75.)

Answer

You can figure the standard deviation using the formula or the steps.
Using the formula: SD = YSD 2 = V3.75 = 1.94.
Using the steps:

0 Figure the variance. The variance (from above) is 3.75.
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.

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

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

Women (n = 60) Men (n = 60)

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.

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: -5, -4, -1, -1, 0, -8, -5, -9, -13, and -24. 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 (SD = 3.18).” 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.

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
(M = 9.21; SD = 7.34) 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

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

Target

Prime

Black White

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.

The 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
o Enter the scores from your distribution in one column of the data window.
@ Analyze.
• Descriptive statistics.
O Frequencies.
• on the variable for which you want to find the mean, mode, and median, and

then _- the arrow.
O Statistics.
#11 Mean, Median, Mode, Continue.
0 Optional: To instruct SPSS not to produce a frequency table, the box labeled

Display frequency tables (this unchecks the box).
O 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

PSYC 354

Excel Homework 3

(70 pts possible)

The objective of your third Excel assignment is to learn to describe a data set using measures of central tendency and variability. First, be sure you view the presentation that covers computing central tendency and variability in Excel found in the Reading & Study folder in Module/Week 3. This presentation goes through the steps you will need to be familiar with in order to complete this assignment.

In Module/Week 3, the goal is to use Excel formulas to calculate specific measures of central tendency and variability of a given data set, using the steps you learned during the presentation. Open “Data Set 3,” found in the Assignment Instructions folder, under “Excel Homework 3,” then follow the steps below to complete Module/Week 3’s assignment.

1.
Research Question:
In Module/Week 3, the data comes from an internet survey that assessed the frequency of use of the social networking site Facebook ™. A psychologist interested in time spent visiting a social networking site collected data from 366 respondents concerning: 1.) number of visits to FB per day; 2.) number of times participants changed FB “status” per week; and 3.) participant age. She is interested in summarizing the data set using measures of central tendency and variability, and displaying these in a table.

2. In the Excel file, each participant’s responses are recorded in columns A through C, rows 2–367. As you scroll down through the data, you may notice that some values are missing. This is intentional and often happens when participants respond to certain items but not others on a survey.

3. You will also see a table to the right of the data which contains labels but no numbers. It is THIS table that you will be completing during this module/week’s assignment. An important note: It is essential to leave data values as they are during this module/week’s assignment—do not change any numbers. Your assignment grade is based on both the numerical values themselves as well as the formulas you use to derive them. Changing a number in the data set will affect the results of any calculations using the formulas (even if only slightly). If you find that you have changed a data value and forgotten it, you can open the original file from the Data Set document, compare values, and make any necessary changes to your own file.

4.
Format Table Cells

Begin by formatting the blank cells within the table as the category “Number” with 2 decimal places. (See the presentation from the Reading & Study folder in Module/Week 1 on formatting cells if you need a refresher.) This will give your table a uniform look. (10 pts)

5.
Compute Measures of Central Tendency

a. Using the “AVERAGE” function, display the means of all three variables in the appropriate cells of the table. (10 pts)

b. Using the “MEDIAN” function, display the medians of all three variables in the appropriate cells of the table. (10 pts)

c. Using the “MODE” function, display the modes of all three variables in the appropriate cells of the table. (10 pts)

6.
Compute Measures of Variability

a. Using the formula that combines the “MIN” and “MAX” functions (as shown in this module/week’s presentation), display the range of all three variables in the appropriate cells of the table. (10 pts)

b. Using the “VARIANCE” function, display the variance of all three variables in the appropriate cells of the table. (10 pts)

c. Using the “STDEV” function, display the standard deviation of all three variables in the appropriate cells of the table. (10 pts)

You can test whether your formulas are properly linked by changing a value in one of the data columns—the numbers in the corresponding table cells should change (though usually only very slightly due to the large number of scores). However, remember to re-enter the original number in the data column as your grade is based on the data set as it was when you opened it in the Data Set document.

Save the completed file to your computer as “yourname_Excel3.xls”. Your finished Excel Worksheet should be submitted by 11:59 p.m. (ET) on Monday of Module/Week 3.

Page 1 of 2

>Sheet

1 Number of

Visits Per Day Number of

Status Changes Per Week Age 5

3 8 Variable Mean Median Mode Range Variance Standard Deviation 15

Visits Per Day
5 1

Status Changes Per Week
25

Age
5

5 25
6

10 2

4 Data from:

http://www.statcrunch.com/5.0/index.php?dataid=

1 10

28 3 3

19 10 1

20 1 1

42 1 0 48
1 0

32 1 1

27 15 1 18
20 7 19
0 0

40 0 0

30 15 5

26 4

17 31 1 2 19
4 1

41 10 3 20
3 0 20
5 1 32
5 3

24 5 1 18
5 1 20
5 6 21
30 1 20
15 1 20
5 3 41
15 5 19

3 3 19

0 0

49 5 4 31
1 1 18
2 1 20
4 2 20
0 0

52 0 0 27
10 10

13 5 3 18
4

5 1 18

2 0 19
4 1 21
2 1 18
1 1 21
8 1 24
20 20 19
10 10 18
50 100 22 30 2 24
5 3 20
50

14 23 3 7 20

5 1 20

1 1

33 3 18
4 15 14
1 0 25
4 1

54 7 3 19
4 0 20
10 0 19
0 0 42
15 3 20
4 1 28

0 0 40

1 1

66 10 3 14
5 2 20
1 2

29 10 5 24
0 0 50
3 2 17
3 1 19
2 0

63 10 0 18
2 1 21
25 7 19
15 2 20
2 0 20
0 0 25
3 3 31

0 0 25
5 1 20

2 1 33
2 0 24
0 0 20
0 0 21
1 0 21

0 0 27

5 0 20
10 3 19
1 1 19
5 7 18
2 1

38 100 0 22

2 1 21

15 3 26
0 0

36 10 10 24
5 0 18
5 1 19

4 2 20

2 1 49
3 1 18
8 4 18
5 1 26
0 1 32
20 10 21
3 1 37
15 5 15
0 0

56 15 0 20
0 0 28
56 56 18
5 1 36
0 0 14
2 1

47 10 5 30
2 2 21

10 1 20

6 1

35 25 1 18

0 0 42

20 8 32

0 0 21

2 2 23
1 1 37
15 5 20
5 1

20 7 19

1 0

62 1 1

58 1 1 20
3 3 21
2 8 24
1 3 28
2 3 38
10 5 40
5 2 21
1 1 12
2 1 13
5 2 26
3 2 37
0 0 35
10 8 54
0 0 13
76

1 18

1 1 20

10 4 19
4 1 20
0 0 48
0

0.5

23
5 10 12
12

58
2 2 19
1 1

39 0 0 33
3 2 19
7 30 25
1 1 48
2 0

53 20 2 22

1 1 27

25 14 47
1 0

43 4 0 50
3 0 19
1 1

11 0 0 45
3 5 28
1 4 24
60

20 19
20 15 29
10 5 19
20 0 20
1 0 14

0 0 52

3 4 26
20 7 18
2 1 26
3 0 32

1 1 20

2 3 18

0 0 40

19
1 0 52
0 1

51 7 2 19
4 2 22
10 7 19

10 7 19

3 0 50

4 1 20

0 0 58
36
2 1 43
1 0 22
5 1 23
7 1 20
21

5 1 19

3 3 26

3 2 19

3 8 32
7 2 13
56
3 0 18

10 5 19

4 2 18
3 2 35
1 1 28
5 3 19

3 1 19

38
10 2 21
8 2 23
10 1 27
42
58
2 0 23
4 3 39
20 4 18
3 2 18

3 2 19

0 0 53
2 2 38
10 5 41
100

4 3 18
10 2 36
20 20 17
10 2 20
4 0 28

50 100 29

2 1 28
6 14 32
5 0 23
2 3 19
10 10 25
1 2 18

2 3 19

8 20
4 2 19
44 28
3 1 29

3 2 18

2 0

34 5 1 43

2 1 43

1 1

2 1 18

10 3 18
15 4 18
5 30 20
25 2 18
25
1 0 40
1 0 26
3 1 20
1 0 19
5 2 24
3 3 25

10 2 20

15 1 19
30 2 20
1 1 41
3 1 17

10 5 19

4 4 24
0 0

2 3 19

20 2 20
12 14 43
0 1
5 1 21

5 3 18
5 1 19

3 0 24
75 7 14
4 2

46 2 0 32

1 0 21

3 1 25
6 2 19
2 0 27

5 1 21
1 0 43
19

12 17 25
1 0 46
2 6 19

1 0 48
1 0 25

8 1 18
1 1

4 34
0 0

55 6 3 20
4 5 20

5 1 18

20 2 44
7 4 18

10 0 18

4 1 19
23
125

62 48

3 2 33
9 5 25
2 1 42
1 0 49
48
10 2 55
1 3 23
1 0 56
13 12 19
1 1 55
0 1 36
50 100 18
1 1 52

1 0 19

1 0 55
2 2 24

1 0 39
2 1
0 0 43
3 0 25

0 0 56
0 1 36

8 1 41

5 1 26
3 0 20

1 0

73 2 3 26
2 2 60
4 15 15
2 1 24
30 3 19
3 0

61 0 0 47

10 8 18
0 0 39

1 0 39
0 0 55
0 0 50

3 1

59 10 3 22
60
2 0 48
40

1 1 35

4 0 20

http://www.statcrunch.com/5.0/index.php?dataid=485373

Sheet2

Sheet3

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