Nursing Home Comparative Data

Data is gathered and compiled by various organizations and often used to inform internal decisions. However, sometimes data are also shared in order to help others make decisions. This week you will be looking into an instance of this data sharing by The Center for Medicare and Medicaid Services (CMS). Specifically, you will be using the the CMS Nursing Home Compare tool to gain insights about the information that is reported to CMS, and in turn how it is presented to the public.

To complete this case study, use the website:

Medicare Nursing Home CompareLinks to an external site.

Instructions to run comparison:

Search for any city/zip code with your assigned state. (See the published announcement for state assignments.) Ensure the “Provider Type” is “Nursing homes”.
Select 3 Nursing Homes from your assigned state using the “Compare” option.
Once you have made your choices, they will appear in a dark blue bar near the top of the screen. Select the Compare button within this row to see data presented in a table for all three selections.
You will be taken to a report page that shows you the ratings for each of the nursing homes scored across various categories.
Scroll down to select the “Quality measures” section.
Locate the following five measures:
Short-Stay –

Percentage of short-stay residents who were re-hospitalized after a nursing home admission.
Percentage of short-stay residents who needed and got a vaccine to prevent pneumonia.

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Long-Stay –

Percentage of long-stay residents experiencing one or more falls with major injury.
Percentage of long-stay high-risk residents with a urinary tract infection.

Select one additional measure of your choice that you would want to review if you had to send a loved one to the facility (this can be from either the short-stay or long-stay category – your choice).

***Note: For some nursing homes, instead of a percentage, you will see a “Not Available” message. In these cases, please select an alternative facility that has all of the data available.

Create an Excel Spreadsheet to demonstrate a comparison between the nursing homes:

Include the names of the three Nursing Homes selected
Compare the Overall rating from the Overview section, not the Quality of Resident Care section, between the three nursing homes. (The rating will be listed out of 5 stars and will be the first star rating at the top of each nursing home.)
Compare the 5 Quality Measures listed above (including the one of your own choosing) between the three nursing homes and also against the State Average and the National Average (included under the name of each individual measure on the website).

Writeup

List the centers you chose for your comparison
Discuss why you selected the fifth quality measure and what makes it important in your evaluation of a nursing home.
Based on your comparison, explain which would be your top choice to place your loved one for a short-stay ANDwhich one you would select for a long-stay. (This must include a discussion of how the selected nursing homes rank in terms of percentages compared to the other nursing homes and to the state and national averages. Use the data from your spreadsheet to support your discussion.)

Formatting Instructions

The case write-up should be formatted using:

Microsoft Word document with 1” margins all around
12 pt. Times New Roman font
Cases should be written in paragraph format, free from grammatical errors.

Submit your Excel spreadsheet and Write-up

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E-mail: J.Eddy@Elsevier.com

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C H A P T E R 2
BASIC MATH CONCEPTS, CENTRAL
TENDENCY, AND

DISPERSION

c0002

clas

cumulative frequen

decimal
dispersion
fraction
frequency
frequency distribution
mean

median
mode
percentage
proportion
quartiles
quotie

range
rate

ratio
relative frequency
rounding
standard deviation
variance
volume

LEARNING OBJECTIVES
At the conclusion of this chapter, you should be able to:
1. Perform calculations with fractions, decimals, and

percentages.
2. Understand the function of rates, ratios, and

proportions in healthcare statistics.

3. Explain why frequencies and frequency distributions
are useful to data analysts.

4. Identify the most useful measure of central tendency
for a given set of data.

5. Calculate the variance and standard deviation from a
frequency distribution.

p005

o00

CHAPTER OUTLINE
FRACTIONS, DECIMALS, AND

PERCENTAGES

Fractions

Decimals

Changing Fractions to
Decimals

Percentages

RATIO, RATE, AND PROPORTION

Ratio

Rate

Proportion

VOLUME, FREQUENCY, AND
FREQUENCY DISTRIBUTION

Volume

Frequency

Frequency Distribution

MEASURES OF CENTRAL
TENDENCY

Mean

Median

Mode

Adjusted Mean

DISPERSION

Interquartile Range

Variance

Standard Deviation

REVIEW QUESTIONS

KEY TERMS

UNIT I Understanding the Basics of Statistics and Data Analytics20

10002-DAVIS-9781455753154

Medicine, like many sciences, uses the metric system to measure weights, lengths,
and fluid volumes. Refer to the inside back cover of this text for a guide to converting
metric and standard units.

b0020

Stat Tip

For some of you, this chapter may be a review that allows you to become reacquainted
with concepts learned in an earlier academic setting. For others, this may be a neces-
sary kick-start to the math required to carry out calculations used in health statistics

and analytics. If you feel that you are solid on a concept, go ahead and try the exercises for
that section. If you find that you are getting them right, by all means, go ahead to the next
section. If not, there is no shame in taking the time to brush up on the areas that you may
not have used for many years. Remember that the phrase “if you do not use it, you lose it”
applies to math concepts as well as everything else in life.

Let us start by looking at some mathematical concepts that you encountered in school a
long time ago.

p00

p0085

BRIEF CASE

UNDERSTANDING THE POPULATION

In the administration of any health care facility, the size and scope of the
patient population help determine the resources needed to deliver care,
like staffing and technology. Part of Sasha’s job is to get a sense of the
kinds of cases the hospital is treating, and the length of time it takes to
treat those cases. To do this, she needs to look at some patient data and
calculate the numbers of patients treated.

b0025

Fractions Numbers that are
expressed as parts of a whole.b0030

FRACTIONS, DECIMALS, AND PERCENTAGES

Fractions, decimals, and percentages are different ways of expressing the same values.
Throughout your experience in health care, you will need to use these numbers to commu-
nicate your findings. Although these concepts are all related, they each often appear sepa-
rately, and you will need to be able to use, calculate, and convert them to their related forms.

Fractions

Fractions are numbers that are expressed as parts of a whole. While we may not always rec-
ognize the use of fractions, they are common in our everyday lives. For example, every Friday
night you might order a large pepperoni pizza. The pizza arrives already cut into eight pieces.
On a normal Friday night, your very hungry roommate eats at least five pieces of that whole
pizza pie. If this was expressed as a fraction, we could say that he ate 5/8’s of the pizza. The
top number (his five pieces, called the numerator) is the parts of the whole that we measured,
and the bottom number (8, called the denominator) is the total (whole) number of pieces.

Other examples include baking (you use a ¾ cup of brown sugar in your chocolate chip
recipe); time (it takes you a half of an hour to walk my dog); parking (it costs a quarter to
buy 12 minutes on a parking meter in the city); shopping (you get 1/3 off when using a cou-
pon from the newspaper); and snow accumulation (we just got 13½ inches of snow). Can
you think of some other examples? Note that each time, the numerator on the top is the
number of parts, while the denominator on the bottom is the total number of parts that the
piece is divided into. When the numerator and the denominator are the same, the fraction
is equal to 1. For example, 4/4 = 1, 70/70 = 1, 14/14 = 1. If your roommate was really hungry
and he ate 8/8 slices of pizza, then he ate one whole pie.

Simple fractions are those that are less a whole number (3/4, 6/7, 9/10), while compound
fractions (also called a mixed number fractions) are those that represent numbers greater than
one (1½, 3¾, 56¼). Compound fractions can also be expressed with a numerator that is larger
than the denominator. These are sometimes called improper or top-heavy fractions. For example,

s0010

p0090

s00

p0095

p0100

p0105

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 2

10002-DAVIS-9781455753154

is the same amount as 1

1
2

15
4

is the same amount as 3

3
4

and

225
4

is the same amount as 5

1
4

You can convert an improper fraction to a mixed number fraction by dividing the
numerator by the denominator to the nearest whole number and showing the amount left
over (i.e., the amount less than one) as a fraction. In the second example above, we ask, how
many 4s can fit into 15 without going over 15? Two 4s would be 8 (2 × 4), three 4s would
be 12 (3 × 4), and four 4s would be 16 (4 × 4). Sixteen is too many, so we know we can fit 3
wholes of this fraction into 15. That leaves ¾ left over. The mixed number fraction 15

4
is the

same thing as saying 33
4
.

We convert mixed number fractions to improper fractions by reversing the process.
Multiply the denominator (in this example, 4) by the whole number (3) to get 12. Then add
the remaining fraction (¾) to get 15/4. Figure 2-1 shows how both of these operations work.

p0110

p0115

4⁄4 goes into 15⁄4 3 times with ¾ left over,

15
4

4
4

3
4

�

� 1 � 1 � 1 �

� � �( ) 4
4( ) 4

)

3
4

Step 1: Multiply the denominator (4) by the whole number (3)
4 � 3 �

Step 2: Add the numerator (3) to the sum of the denominator
and whole number (12)
12 � 3 � 15 This is the new numerator (15)

Step 3: Keep the same denominator (4)

Answer:

3
4

�

3
43

�

15
4

3
4

�

15
4

3
4

�

Figure 2-1 Converting fractions.
f0010

[AU1]

UNIT I Understanding the Basics of Statistics and Data Analytics

MATH REVIEW

If my roommate eats 5/8
of the pizza, and I eat
1/8, did we together eat
6/16?

No, we ate 6/8, or ¾ of the
pie.

b00

ICU Intensive care unit.b00

What are some examples of fractions in health care? We can use them any time we are
working with parts of a whole. If there are 10 beds in the intensive care unit (ICU), and
seven are filled, our fraction is 7/10. If we had 1000 discharges last year, and 13 of those
patients had hospital-acquired pneumonia (meaning they contracted the disease while they
were in the hospital), then the fractional representation is 13/1000.

Frequently, we reduce fractions to make them easier to understand and to work with.
If 10 of the 20 cribs in the nursery are full, we probably would not say the nursery is 10/20
(ten-twentieths) full. We would reduce the fraction to ½, and we would say that it is half-
full. This works because of one of the neat things you can do with fractions: when you mul-
tiply or divide the numerator and the denominator by the same number (called a factor), it
does not change the value of the fraction. For instance, consider the following:

10
20

÷ 10=
10÷ 10
20÷ 10

=
1
2

15
20

÷ 5=
15÷ 5
20÷ 5

=
3
4

÷ 8=
16÷

64÷ 8

=
2
8

÷ 2=
2÷ 2
8÷ 2

=
1
4

Notice that in the last example, we did not reduce the fraction all the way to its simplest
form the first time when we divided by 8. We could have skipped a step and divided 16/64
by 16 and still arrived at the same simplest fraction, ¼.

Multiplication works exactly the same way—we can multiply the fraction by whatever
factor we want, as long as we do the same thing to both the numerator and the denominator.

2
3

× 10=
2× 10
3× 10

=
20
30

Changing fractions by multiplying and dividing is important because if we want to add
or subtract them, the denominator has to be the same number. And we do not add the
denominators, because that is just the total possible.

Let us say the medical-surgical (med-surg) unit
on the second floor is 5/12 full, and the med-surg
unit on the third floor is 1/12 full. Workers on the
3rd floor need to shut off the air conditioning for
repairs, and the hospital decides to move (add) the
patients from the 3rd floor to the 2nd floor. How
many patients will be on the 2nd floor med-surg unit
after the patients are moved? In this case, the addi-
tion is easy, because the denominators are the same.

5
12

+
1

=
6
12

=
1
2

After the move, the unit on the 2nd floor will
have 6 patients, and since there are 12 beds, it will
be ½ full. But let us try adding fractions where the

denominator is not the same. Say the 3rd floor was 1/3 full, and the 2nd floor is ½ full. Will
there be enough room on the 2nd floor? (Note: 1/3 + ½ does not equal 2/5!)

p0120

p0125

p0130

p01

p0140

p0145

p01

Did you know that the word fraction is derived from the Latin word fractus meaning bro-
ken? Fractures are broken bones, while fractions are numbers that are broken into parts.b0035

Stat Tip

f0050

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2

10002-DAVIS-9781455753154

To add (or subtract) fractions with different denominators, we must multiply the frac-
tions by some factor first so that we are adding fractions with the same denominator.
Remember, we can multiply or divide a fraction any way we want without changing its
value, as long as we treat the denominator and the numerator the same.

1
3

× 2=

2
6

Patients from the 3rd floor

1
2

× 3=
3
6
Patients on the 2nd floor

2
6

+
3
6

=
5
6
Adding them together, the 2nd floor will be 5/6 full aer the patients are moved.

p0155

1. Convert the following improper fractions to mixed number fractions:

a. 12
8

b. 5
2

c.
1

2. Convert the following mixed number fractions to improper fractions:

a. 3
3
8

1
2

5
16

3. Reduce the fractions below to their simplest form.

a. 2
8

b. 50
100

c. 75
1000

d.
12
144

e.
6
36

b0050

EXERCISE 2-1
Fractions.

UNIT I Understanding the Basics of Statistics and Data Analytics

10002-DAVIS-9781455753154

Decimal A fraction with a
denominator based on the
number 10.

b00

Decimals

Decimals are related to fractions in that they are numbers that are divided into units of 10.
Decimals are actually fractions whose denominators are some power of 10 (10, 100, 1000,
etc.) and are written as a decimal point followed by the numerator. For example, the frac-
tion 1/10 (one-tenth) can be written as the decimal 0.1. 764⁄100 (Seven and sixty-four one-
hundredths) is expressed as 7.64 in decimal form. Again, these numbers, like fractions, are
describing parts of a whole. The difference between fractions and decimals is not only in the
way they look, but also in the concept of a whole. In decimals, the whole is always divisible by
10 (for example: 10, 100, 1000). The decimal point separates the whole from the parts (like the
line between numerator and denominator), but in decimals, the whole numbers are to the left
of the decimal point, while the parts are to the right. Figure 2-2 shows the numbers that each
of the placeholders represents, along with its notation and the prefixes associated with each.

Changing Fractions to Decimals
Sometimes, you will need to change a fraction into a decimal for performing a calcula-
tion. A great example is the sale coupon that gives you 1/3 off of your purchase. One of the
t-shirts that would be perfect for my niece (and my budget) is priced at $36.00. If it is 1/3
off, how much will I save? I might first think to divide the price into three’s, then multiply
by two. A third off would be the following:

s0020

p0160

s0025
p0165

BRIEF CASE

WORKING WITH FRACTIONS

One of the clinics attached to the hospital system handles walk-ins
and provides some urgent care services. Of the 120 patients seen last
month, 10 were Asian-American, 35 were Latino or Hispanic, and 15
were African-American. Sasha wants to report these ethnicities in sim-
ple fractions.
Determine the fraction of the whole for each ethnic group and report

your findings in simple fractions.
Asian-American:
Latino or Hispanic:
African-American:

b0055

Decimals, like fractions,
describe parts of a whole.b0065

TAKE AWAY l

4. Add or subtract the following fractions. Report your answers in simple fractions.

a. 1
8

+
7
12

b. 7
8

−
1
16

c. 1
2

−
1
5

d. 2
3

+ 1
1
3

e. 3
5
8

+ 7
3
4

EXERCISE 2-1
Fractions.—cont’d

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 25

10002-DAVIS-9781455753154

Quotient The result of division. b00

36
3

= 12× 2= $24. I saved $12.

But, you can also change the 1/3 into a decimal, then multiply it by the price to see how
much you are going to save. Let us say a second, equally enticing t-shirt is 2/5 off and is
priced at $39. $39 is not easily divisible by 5, so making that fraction (2/5) into a decimal
might be easier. To convert, we just divide the numerator (which is 2) by the denominator
(5) to get the quotient.

2÷ 5=

0.4

$39× 0.4 = $15.60

$39.00− $15.60 = $23.40

Since I did the math, I can see that the second t-shirt is actually cheaper, even though the
original price was higher.

Changing the fraction to a decimal leads us to another important concept: rounding.
Rounding is a method of reducing the number of digits in a number so that it is less precise,
but is more convenient to use. For example, to change a fraction to a decimal, you divide
the numerator by the denominator. 2/5 = 0.4, four tenths. That is a pretty easy number to
work with. However, dividing 1/3 gives us a quotient of 0.3333333333… and on forever.
The 3s just keep repeating infinitely. To come up with a usable number, rounding rules need
to be applied. To round a number to one decimal place (like 0.4), you look at the number
immediately to the right of the place holder that you want to round to. If the number is
between 0 and 4, you drop the remaining digits and leave the number in the tenths place as
it is. This is called rounding down. If the digit is between 5 and 9, you add one to the digit
in the tenths place. This is called rounding up. In this case, the number in the 100ths place
is a 3. Three is between 0 and 4, so you leave the 3 in the tenths place alone. The rounding
process results in a 0.3. Although rounding leaves you with a number that is not as precise
as your original result (0.333333333), it allows you to perform calculations that would be
difficult, if not impossible.

In many health care applications, converting to a decimal makes a fraction easier to use.
Let us say the city of Midville, Florida has three hospitals—two are larger facilities, and one
is smaller. Of all the admissions last year, very few patients had a principal diagnosis of
MRSA, a kind of bacterial infection that is difficult to treat with antibiotics.

Facility 2015 Admissions 2015 MRSA Cases
Midville General Hospital 14,065 2
Midville Lutheran Hospital 4,023 1
University of Midville Medical Center 12,200 1

p0170

p0175

p0180

p0185

t0040

Place Value and Decimals

4 8 2 9 . 1 7

M
ill

io
ns

H
un

dr
ed

th
ou

sa
nd

T
en

th
ou

sa
nd

T
ho

us
an

H
un

dr
ed

T
en

O
ne

D
ec

im
al

p
oi

T
en

th
s

H
un

dr
ed

th
s

T
ho

us
an

T
en

-t
ho

us
an

dt
hs

H
un

dr
ed

-t
ho

us
an

dt
hs

M
ill

io
nt

Figure 2-2 Decimal placeholders.
f0015

Convert fractions into deci-
mals by dividing the numera-
tor by the denominator.

b0075

TAKE AWAY l

Rounding Reducing the number
of digits in a number to make it
easier to use.

b0080

UNIT I Understanding the Basics of Statistics and Data Analytics26

10002-DAVIS-9781455753154

Percentage The number of times a
thing occurs out of 100.b0095

MATH REVIEW

If we say 0.00047 patients of
all the patients in Midville had
MRSA, how many people is that?

The 4 is in the 10,000s place,
so we might say 4.7 infections for
every 10,000 people. Or, we might
say 47 of every 100,000 patients
were treated for this infection.

b0085

What fraction (part of the whole) of patients in all three Midville hospitals had MRSA? We

know how to set up the fraction for each:
2

14, 065
+

1
4, 203

+
1

12, 200
. But we would not

want to try to find the common denominator of all these fractions in order to add them
together. It would be much easier (though slightly less precise) to convert each fraction to a
decimal, then add the decimals. Let us look at the math:

p0190

Facility
Numerator
(MRSA Cases)

Denominator
(Total Patients) Quotient Rounding

Midville General Hospital 2 ÷ 14,065 = 0.00014219694 0.00014
Midville Lutheran Hospital 1 ÷ 4023 = 0.00024857072 0.00025
University of Midville Medical Center 1 ÷ 12,200 = 0.00008196721 0.00008
Total 4 ÷ 30,288 = 0.00047273487 0.00047

t0045

Calculating the part of the whole of the patients in Midville who were treated for MRSA
using fractions would be difficult; but when we convert the fractions to decimals and use
rounding, we can see that 0.00047 of all the patients (30,288) in Midville had MRSA.

p0195

1. Convert the following fractions to decimals:

a. 3
8

b. 13
1
2

c. 7
5
16

d. 1
160

e. 60
10000

2. In the decimal 0.012358467, the digit 1 is in the __________ place.
3. In the decimal 0.193847, the digit 7 is in the __________ place.
4. Round each decimal to the tenths place. Then round each to the hundredths place.

Then round to the thousandth place.
a. 0.09513999
b. 0.5510

c. 1.342809

b0090

EXERCISE 2-2
Decimals.

Percentages

Like a decimal, a percentage is also based on the number 10, or more precisely, the number
100. A percentage is the number of times something occurs out every 100 times. Percent-
ages are useful because often, just stating the amount of something is confusing to the user.
Presentation of the percentage standardizes the data so that unlike groups can be compared.
One familiar example is the quiz grades you received for a class. If you answered 24 of 27
questions correctly on one quiz, and 30 of 35 questions right on another, which quiz did
you score better on?

s0030

p0200

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2

10002-DAVIS-9781455753154

To calculate a percentage, divide the observations in the category by the total observa-
tions, and multiply by 100.

observations
total observations

× 100 = percentage

Quiz
correct answers
total answers = × 100 Percent

Quiz A 24
27

0.8888888889 × 100 89%

Quiz B 30
35

0.8571428571 × 100 86%

Since we standardized the data by looking at each score out of 100, we can see that the score
on the first quiz was slightly better.

Now let us look at a simple health care application. Consider the question: How many
male and female patients were discharged in February this year compared to last year? We
can look at the difference (the variance) between the number of women and men in each
period, as illustrated in Table 2-1, but the result may not be helpful.

p0205

t0050

p0210

p0215

TABLE 2-1

VARIANCE BETWEEN NUMBER OF WOMEN AND MEN DISCHARGED IN FEBRUARY

FEBRUARY 2015 FEBRUARY 2014 VARIANCE

Males discharged 413 386 27
Females discharged 385 349 36
TOTAL 798 735 63

t0010

Just by looking at the table, we can see that more men than women were discharged in
February both this year and last year. We can also see that discharges for both women and
men have increased. It appears that there has been a larger increase in discharges of women
(36) than in men (27). But is that true? To give a more accurate analysis of the activity in
Table 2-1, we should also provide the percentage of observations and the percent variance.

For example, if we want to know what percent of the patients discharged in February
2014 were women, we would do the following:

349 women discharged
735 total discharges

= 0.48246× 100= 48%

The calculation shows that 48% of the discharges in February 2014 were women. But
what about the variance? How many more women were discharged in 2015? We can use
the same calculation (observations divided by total observations) to determine the percent
variance, showing exactly what happened:

variance of women discharged
total women discharged

=
36
349

= 0.1032× 100= 0%

Table 2-2 expands the data to include the percentages. Now, it is clear that the total
number of discharges has increased by 8.6%, the percentage of women increased by 10.3%,
and the percentage of men increased by 7.0%. These are descriptive statistics, so we can-
not say why there is a greater percentage increase in women patients than in men. We will
have to examine this data over a longer period of time and look further into the types of
illnesses and treatments that the patients have to understand the reason for the change, if
it continues.

p0220

p0225

p0230

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UNIT I Understanding the Basics of Statistics and Data Analytics

10002-DAVIS-9781455753154

Calculating percentages can
allow you to estimate the
impact of a decrease (or
increase) in patient volume,
which can tell you the num-
ber of personnel needed for
a particular medical service.

b0100

TAKE AWAY l

Fractions, decimals, and percentages are closely related concepts, and in practice, you
will need to be able to convert between these formats frequently. Box 2-1 summarizes the
relationships between these concepts.

p0240

TABLE 2-2

PERCENTAGE CHANGE IN NUMBER OF WOMEN AND MEN DISCHARGED IN FEBRUARY

PATIENTS
DISCHARGED

FEBRUARY 2015 FEBRUARY 2014

VARIANCE % VARIANCENUMBER % OF TOTAL NUMBER % OF TOTAL

Male 413 52% 386 53% 27 7.0%
Female 385 48% 349 47% 36 10.3%
TOTAL 798 100% 735 100% 63 8.6%

t0015

RELATIONSHIPS BETWEEN FRACTIONS, DECIMALS, AND PERCENTAGES

FRACTION DECIMAL PERCENTAGE

1/100 0.01 1%

5/100, 1/20 0.05 5%

10/100, 1/10 0.1 10%

1/8 0.125 12.5%

25/100,5/25, 1/5 0.25 25%

50/100, 1/2 0.50 50%

100/100, 1 1.0 100%

125/100 1.25 125%

200/100 2.0 200%

TO CONVERT FROM A FRACTION TO A DECIMAL
Divide the numerator (top number) by the denominator (bottom number).

TO CONVERT FROM A DECIMAL TO A FRACTION
Divide the decimal number by the power of 10 that it represents, then simplify the frac-
tion.

TO CONVERT FROM A DECIMAL TO A PERCENT
Multiply the decimal by 100, and add a percentage sign.

TO CONVERT FROM A PERCENT TO A DECIMAL
Divide the percentage by 100, and drop the percentage sign.

TO CONVERT FROM A FRACTION TO A PERCENTAGE
Divide the numerator by the denominator, then multiply the result by 100, and add a
percentage sign.

TO CONVERT FROM A PERCENTAGE TO A FRACTION
Drop the percentage sign, then divide the decimal number by the power of 10 that it
represents.

b0010 BOX
2-1

Dividing the percentage by 100 results in moving the decimal point two places to
the left. Once you are comfortable with this method, you may want to use it as a
shortcut for conversions.

b0015
Stat Tip

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2

10002-DAVIS-9781455753154

RATIO, RATE, AND PROPORTION

Ratio

While fractions and decimals help to describe parts of numbers that are generally smaller
than one, a ratio is more useful in comparing one group of numbers to another. Some of the
vocabulary and concepts that you have learned about them can be recycled here to explain
this statistical term.

A ratio is a comparison of two or more numbers using the same unit of measurement
(time/dollars/weight). Ratios can be expressed with either a colon or a slash between the
two numbers. As always, an example helps make this concept more understandable. One
common ratio we use in the classroom is the ratio of students to teachers. Let us say in an
average class, there are 23 students for every teacher. The ratio of students to teachers, then,
is 23 to 1, or 23:1.

Another example is the number of nurses to patients on a unit. If a particular floor is
very busy and the nurses are short-staffed, we might see a ratio of 1:6, or one nurse to every
six patients. In the ICU where nursing care is critical, the facility would aim for a ratio of
1:2 or even 1:1 nurse per patient.

Hospital A treated a total of 47 patients at its Saturday morning clinic. Thirty-two of the
patients were female, while 15 were male. The ratio of females to males was 32:15 or 32/15.
This is useful in giving an overall impression of the magnitude of the similarity or differ-
ences between the groups compared; in this case, we can see that there were more women
treated this Saturday—a lot more! Using your knowledge of rounding and reducing frac-
tions, a 30:15 ratio would be expressed as 2:1 in its simplest form.

When working with a ratio, pay close attention to the order. 32:15 is not the same as
15:32! Also, if the units are of different magnitudes (example: minutes vs. hours for time),
you need to convert them to be the same unit to make a comparison. Let us say we are look-
ing at two people, Althea and Zinnia, who are both coding medical records. If Althea codes
6 records every 30 minutes (6:30), then reducing the ratio we could say she codes 1 record
every 5 minutes, or 1:5. Zinnia, on the other hand, codes 13 records an hour, or 13:1. Who is
faster? We would have to convert Zinnia’s magnitude to minutes—or Althea’s to hours—to
find out.

s0035

s0040

p0245

p0250

p0255

p0260

p0265

1. Calculate the following percentages
a. 10/50
b. 49/100
c. 17/1000
d. 14/16
e. 1810/2000
2. Convert the following percentages to decimals.
a. 1%
b. 10%
c. 47%
d. 0.5%
3. Convert the following to the simplest fraction.
a. .5
b. 0.98
c. 0.333333333
d. 1.75
e. 90%
f. 25%

b0105

EXERCISE 2-3
Percentages.

MATH REVIEW

There are 60 minutes in 1 hour. Al-
thea’s ratio is 6 records to 30 min-
utes (6:30), so we can multiply that
by 2, finding she codes 12 records
every 60 minutes. Zinnia’s coding
ratio is 13 records to 60 minutes
(13:60), so she is slightly faster.

b0120

Ratio A comparison of two or more
numbers using the same unit of
measurement.

b0110

ICU Intensive care unit.
b0115
[AU2]

UNIT I Understanding the Basics of Statistics and Data Analytics30

10002-DAVIS-9781455753154

Proportion The relation of four
quantities in two equal ratios,
where the first quantity divided
by the second equals the third
divided by the fourth.

b0140

DRG Diagnosis-related group.b0135

Rate A value in relation to a
different unit.b0125

MATH REVIEW

If you can do one example every
3 1⁄3 minutes, how long does it
take to do 12? 12 examples × 3
1⁄3 min = 40 minutes.

b0130

Rate

Rates are comparisons of two numbers that are measured with different units of measure-
ment, calculated as a numerator divided by a denominator. To make an example that is
close to your study, you might want to calculate how many minutes it takes you to work
through examples in the book. If you can do three examples every 10 minutes (or three
examples/10 minutes), you can use the calculation to realize it takes you about 31⁄3 minutes
per example. If you know that you have 12 examples to work through, you can use the rate
to find out how many minutes the exercises will take.

Rates need to be labeled with the unit that is being measured. In this example,
the unit would be “minutes per example.” If we were comparing the dollars in DRG
(diagnosis-related group) reimbursement for each DRG, the unit would be “dollars per
case.” If we looked at the number of malpractice claims in the hospital by month, it
would be claims/month. Note that this is different than ratios that are expressed as
simply one number compared to another, because in ratios, the units of measure are
the same.

Rates are very, very common in health statistics and data analytics. Because there is often
an element of time involved, facilities frequently use rates to determine how they are per-
forming. For any given month or year, they might examine birth rates, death rates, rates of
infection, rates the physicians consulted with one another—just to name a few. We will look
at the kind of rates used for benchmarking more closely in Chapter 5.

Proportion

Proportions are expressed as two equal ratios. ½ cup = 2/4 cups. Proportions are useful in
figuring out unknown values when you know one ratio and want to determine what one of
the other two variables would be.

Let us say that it is a hospital policy to make sure there are always three nurses for every
patient on the med-surg floor. This hospital has found that if the nurses become more out-
numbered than that (e.g., 4:1), the quality of care suffers, and if there are fewer than that
(e.g., 2:1), the nurses have quite a bit of downtime. So, if today we have 24 patients and we
want to make sure we have the right amount of nurses, how do we set up the proportion? In
this case, some basic algebra helps us solve the problem.

1 nurse
3 patients

=
x nurses

24 patients

To solve the problem, you can use a trick called cross-multiplication:

=
3 patients 24 patients
1 nurse x nurse

3x= 24

x= 8

To meet quality standards, we would have to staff this floor with 8 nurses.
Proportions can be either direct or inverse. A direct proportion is one in which there

is an increase in one quantity when there is an increase in the other—or a decrease in one
quantity when there is a decrease in another. We might observe that as the number of nurses
per floor increases, so does the number of positive comments per satisfaction survey. This is
a direct proportion. An inverse proportion is the opposite: when there is an increase in one
quantity, there is a decrease in the other. As the number of flu shots per month increases,
for instance, the number of cases of flu per month decreases.

Table 2-3 compares rate, ratio, and proportion.

s0045

p0270

p0275

p0280

s0050

p0285

p0290

p0295

f0060

p0300
p0305

p0310

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 31

10002-DAVIS-9781455753154

VOLUME, FREQUENCY, AND FREQUENCY DISTRIBUTION

In the last chapter, we talked about how descriptive statistics are used to give an overall
impression of a group of data. As an HIM professional, you may be required to (numer-
ically) describe patients, employees, diseases, procedures, or any number of health care
events. The descriptions may include ages, salaries, outcomes, and many other values. Very
simply, those descriptions will be answers to a series of questions:
How many are there?
What are the characteristics of the group?
How similar/different are the subgroups of characteristics?
What are the relationships between the subgroups?

The individual descriptions for each question (100 patients; 10 are 4 years old; 6 are
37 years old; we pay graduate nurses $45,000/year; 4 patients deceased last month) are the
values that can answer the real question: What is the variable for this particular observation?
The first sets of descriptive statistics that we cover will help answer those questions.

The very first, and simplest, numeric description of a group of data is its volume, or total,
and it is one of the most common questions asked by managers and administrators. They
want to know: how many?
• The hospital admitted 7593 patients last year.
• There were 162 tonsillectomies performed over the last 5 years.
• A physician practice currently employs 26 professionals.

s0055

p0315

u0080
u0085
u0090
u0095
p0340

p0345

u0100
u0105
u0110

1. Alana can code 8 charts in 1 hour. What is the ratio of minutes to charts?
2. The group practice’s policy is that for every 3 physicians in the group, there should

be one medical assistant (MA). If there are 12 physicians in the group, how many
MAs should the practice employ?

3. A radiology center on the east side of town performs 12 X-rays a day. A larger,
competing center on the west side of town performs 30 X-rays a day. What is the
simplest ratio of east side X-rays performed to west side X-rays?

b0145

EXERCISE 2-4
Ratio, Rate, and Proportion.

TABLE 2-3

COMPARISON OF RATIO, RATE, AND PROPORTION

STATISTIC
UNITS OF
MEASUREMENT APPEARANCE EXAMPLE

Rate Different Expressed as a quotient or
with a colon between
the two variables

A family physician sees 3
patients every hour.

3:1 hour or
3patients
1hour

Ratio Same Expressed as a quotient or
with a colon between
the two variables

7 of every 10 patients the physi-
cian sees are over the age
of 60.

7:10 patients or
7 patients over age 60

10 patients
Proportion Different Expressed as two ratios A medical practice has 4 PAs

and 9 doctors. The propor-
tion is

4 PAs

13 clinicians

=
9 doctors

13 clinicians

t0020

UNIT I Understanding the Basics of Statistics and Data Analytics32

10002-DAVIS-9781455753154

Volume The count of an activity
or value.b0150

Notice that these are totals for each example. To determine the total number of patients
treated, for example, an administrator would ask how many admissions or discharges
occurred during a certain period.

Volume

Some questions that ask how many are asking about volume, the count of an activity or
value. Volume is an important measure for defining activity and for comparing activity
from one period to the next or among departments or facilities. It can tell us a lot about
how much of an activity we are doing, or how much more (or less) we are doing compared
to other times or other facilities.

Say, for example, you and the kid next door decide to operate a lemonade stand. At the
end of the month, you count the money the stand collected, totaling $500. This is the vol-
ume for July, describing the total activity for the month. After toasting to your success with
an ice-cold, fresh-squeezed lemonade, you find out that the stand two blocks away made
$700 in July. Calculating volume tells us how much activity we did, and comparing volumes
tells us how much activity we might have done.

In health care, we can provide volume figures on any type of data by counting the total
number of observations of the particular data element, or by counting the number of obser-
vations in a particular category. So, we can count the revenues collected, the number of
female patients, the number of Asian patients, the number of patients who were age 65,
the number of tonsillectomies, and the number of patients discharged on a particular date.

Say the manager asks, “What was the volume of discharges last month?” She wants to
know the total number of discharges (how many) occurred. But to answer a volume ques-
tion, we need to know more about what specific volume is being requested. In particular,
we need to know the following:
1. The month: the first and last dates, including the year
2. The specific type of patient (inpatient or outpatient)
3. If there are any services that must be included or excluded (e.g., emergency department

or same day surgery; newborns or adults)
Say that the manager answers those questions like this: I want to know the volume of dis-

charges in September of 2015 for all adult inpatients. Now we know that we need to count the
number of discharges of adult inpatients that occurred from 9/1/2015 through 9/30/2015.
The next question is what constitutes an adult? Let us say that our hospital considers all
patients over the age of 16 as adults. There are two steps to this analysis. We need to count
only September, 2015, and we need to omit any patients under the age of 16. So, we can
sort the database by discharge date, then sort the month of September by patient age. Then,
count the number of patients who are age 17 and older.

In an electronic record environment, the software has usually been programmed to
report the most common volumes needed for management and administrative purposes.
Some common volumes include number of discharges by date, diagnosis, procedure,
attending physician, and surgeon. Often, variations on those volumes can be queried if
there is no standard report available.

p0365

s0060

p0370

p0375

p0380

p0385

o0300
o0305
o0310

p0405

p0410

Counting can be done manually. It can also be done by using a formula in an electronic
spreadsheet. In Excel, the counting formula is = count(range) for numerical values,
such as counting the number of admission type codes on a list, and = counta(range) for
alphanumeric values, such as counting the number of names on a list.

b0155

Stat Tip

Frequency

Where volume tells us the amount of an activity, frequency describes the number of times a
specific value for a variable occurs. If you wanted to know how many people in the class are
male, for example, you would be asking about the frequency of men in the group. Looking

s0065

p0415
Frequency The total number of

occurrences of a value.b0160

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 33

10002-DAVIS-9781455753154

at the class roster in Figure 2-3 as a sample, the variable is gender, the value is male, and the
frequency is 6 times in this class of 23.

Statisticians sometimes talk about an absolute frequency, the total number of values for
the variable being measured. In the class roster example, the absolute frequency is all 23
students. Relative frequency is calculated as a percentage and is described with percent-
ages. The relative frequency is the observed frequency of a value divided by the absolute
frequency. In other words, it is the ratio of the number of values in a particular category to
the total number of values in that group. It can be described also as a proportion, a part of
a whole, or as a coin toss. Let us look again at our class roster. How can you tell the rela-
tive frequency of males in the class? It is the observed frequency, which is 6, divided by the
absolute frequency, 23, or 26%.

Frequency Distribution

Sometimes answering the question how many does not give the user enough meaning-
ful information. For example, the instructor of this course might think that most of her
students have earned a passing grade. She could just count how many students got a 70,
how many got a 71, how many got an 80, etc. That would give us the volume of students
who received a particular score. However, that just gives us a long list of data that is not
meaningful.

Remember that grades and salaries and ages are examples of continuous, ratio data.
Although they most commonly appear as a whole number (e.g., 90, $45,000, 7 years old),
they are, nevertheless, continuous. A common way to analyze age and other continuous
data is with a frequency distribution. Frequency distributions can include either grouped

p0420

s0070

p0425

p0430

Alex 19 Male

Amanda 21 Female

Ashley 19 Female

Brittany 20 Female

Elizabeth 19 Female

Emily 21 Female

Hannah 23 Female

Jack 20 Male

Jessica 20 Female

John 33 Male

Kayla 19 Female

Lauren 20 Female

Margaret 27 Female

Marion 19 Female

Megan 21 Female

Rick 26 Male

Rob 19 Male

Samantha 22 Female

Sarah 20 Female

Scott 25 Male

Stephanie 22 Female

Sue 29 Female

Taylor 22 Female

Class Roster

GenderAgeName

Figure 2-3 Class roster.
f0020

Relative Frequency The observed
frequency of a value divided
by the absolute frequency (the
total).

b0165

Frequency Distribution The
organization of data into tabular
format using mutually exclusive
classes and frequencies.

b0170

UNIT I Understanding the Basics of Statistics and Data Analytics

10002-DAVIS-9781455753154

or ungrouped data. A grouped frequency distribution takes the categories of the variable
and groups those categories into equal ranges. Each of these smaller groupings of data is
called a class.

Each class must be mutually exclusive, meaning that any value that is assigned to a class
can fit in one and only one class. The class limits (upper and lower) are the values that
separate one class from another. For example, course grades are traditionally divided into
groups or classes of A, B, C, D, and F. The class limits for each are A (90–100), B (80–89),
C (70–79), D (60–69) and F (<60). The upper and lower limits together are called a class interval. Look at Table 2-4. Notice that there are no overlaps in any of the classes—they are mutually exclusive. An 80 goes in the B class and cannot be categorized as an A or a C. If the groups were A (90–100), B (80–90), C (70–80), D (60–70), and F (0–60), you have group- ings that overlap, and an 80 could be either a B or a C. Obviously, that system just will not be acceptable for grades and certainly not for categorizing any type of data.

With each student’s grade grouped into this frequency distribution, the instructor can
see that yes, most students did earn a passing grade. Out of the 23 students in the class,
eight received a C, another eight got a B, and five got As. Since 8 + 8 + 5 = 21, 21 of the 23
students passed.

Let us look at an example in health care. How can a frequency distribution answer the
how many question better than volume? The nursing managers may be telling adminis-
tration that there are a lot of patients who require interpreters. Or that lately they have
too many geriatric patients, putting pressure on the nursing staff to expand the number of
nurses with geriatric competency and challenging the facility’s resources. Now, we could
just count how many patients there were of each age—how many 1 year olds, how many
2 year olds, etc. But again, that would leave us with a long, unhelpful list of data; grouping
the patients into age ranges would make it much easier to see how much of the hospital’s
resources are being utilized to treat certain patients. If the ages of our patients range from 1
to 100, we can group those ages into five ranges of 20 ages each, 10 ranges of 10 ages each,
or 20 ranges of five ages each, as follows:

5 Ranges of 20 Ages 10 Ranges of 10 Ages 20 Ranges of 5 Ages
0–20 0–10 0–5

21–40 11–20 6–10
41–60 21–30 11–15
61–80 31–40 16–20

81–100 41–50 21–25
51–60 26–30
61–70 31–35
71–80 36–40
81–90 41–45

91–100 46–50
51–55
56–60

p0435

p0440

p0445

t0055

TABLE 2-4

CLASS GRADES IN A FREQUENCY DISTRIBUTION

CLASS
LOWER
LIMIT

UPPER
LIMIT

CLASS
INTERVAL

CLASS WIDTH
(OR SIZE) VALUES FREQUENCY

A 90 100 90–100 11 90, 90, 93, 97, 100 5
B 80 89 80–89 10 81, 83, 85, 85, 86,

88, 88, 89
8

C 70 79 70–79 10 70, 71, 73, 75, 75,
78, 78, 79

D 60 69 60–69 10 62 1
F 0 59 0–59 60 49 1

t0025

Class A group of values in a
frequency distribution.b0175

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 35

10002-DAVIS-9781455753154

5 Ranges of 20 Ages 10 Ranges of 10 Ages 20 Ranges of 5 Ages
61–65
66–70
71–75
76–80
81–85
86–90
91–95

96–100

Which grouping is best? It depends on what we are trying to determine. Let us think about
why we are doing this analysis. At the moment, we just want to get a sense of the ages of our
patients, to answer the question where is our concentration of patients? For that purpose, we
can use the five ranges of 20 ages grouping. In the table below, the volume column is the
count of patients in each age range, and the cumulative frequency is a running total of all
classes.

Age Volume Cumulative Frequency Relative Frequency
0-20 78 78 78/1095 × 100 = 7.1%

21-40 173 (78 + 173) = 251 251/1095 × 100 = 22.9%
41-60 251 (78 + 173 + 251) = 502 502/1095 × 100 = 45.8%
61-80 265 (78 + 173+251 + 502) = 767 767/1095 × 100 = 70.0%
81-100 328 (78 + 173 + 251 + 502 + 767) = 1095 1095/1095 × 100 = 100%

What does this distribution tell us? We see very few patients under the age of 20. Over
half of our patients are over the age of 60. So, in terms of hospital services, it seems as
though we have been concentrating on the elderly population. Further analysis is necessary
to determine whether increasing services would be helpful. We would need to look at the
competition in the marketplace as well as the demographic profile of the catchment area
(the geographic area that the hospital serves).

p0450

t0060

p0455

A frequency distribution
may be used to express
volume for a variable that
represents continuous data.

b0180

TAKE AWAY l

MEASURES OF CENTRAL TENDENCY

Measures of central tendency can be remembered as the 3 M’s: mean, median and mode.
You probably recognize the term mean as being synonymous with the word average. Your
average for the course determines your grade; you may decide to browse careers on their
average salary; and everyone wants to be above average. But do you know what an average
is and how to calculate one? Each of the measures of central tendency aims to find a single
value that best represents the rest of the data. Do you know when it makes sense to use the
mean to describe your data and when you should use one of those other M’s?

Mean

The mean is the sum of the values in the data that you are measuring divided by the total
number of observations. Synonyms for the mean are the average, the arithmetic mean,

s0075

p0460

s0080

p0465

1. The children’s wing has 25 male patients and 16 female patients. What is the abso-
lute frequency? What is the relative frequency of males? Of females? What is the
ratio of boys to girls in its simplest form?

2. While looking at salaries of nurses at the hospital, you find a range from a low of
$25,000 to a high of $80,000. How many classes would you have if you broke them
into class widths of $5000?

b0185

EXERCISE 2-5

Mean The sum of the values
divided by the total number of
observations.

b0190

UNIT I Understanding the Basics of Statistics and Data Analytics36

10002-DAVIS-9781455753154

LOS Length of stay.b0215

ALOS Average length of stay.b0210

Outlier An extreme value in a set
of data.b0195

and the expected value. This calculation helps to answer the question: what is the usual
number or amount? For example, you have one course that has 5 exams. You earn a 90,
a 0, a 90, an 80, and a 100. The average for the course is 90 + 0 + 90 + 80 + 100 ÷ 5, and
360 ÷ 5 = 72.

In any group of data, there is only one mean, and that calculation can be affected by
extreme values, called outliers. The exam on which you scored a zero (the outlier) has
a huge impact on your average. Students sometimes are unaware of the effect of an out-
lier. If you had gotten another 80 instead of a zero, your average would have been
90 + 80 + 90 + 80 + 100 ÷ 5, and 440 ÷ 5 = 88.

p0470

To calculate the mean, add
the sum of the group of
numbers, and divide the
sum by the number of items
in the group.

b0200

TAKE AWAY l

Outliers certainly influence important statistics like average length of stay (ALOS) where
most patients stay 2–3 days, but a few stay up to 90 days, greatly skewing the mean. Facilities
regularly use ALOS to determine the amount of resources their patients require. We will
examine ALOS in greater detail in Chapter 4 on administrative data, but here is a simple
example. The table below lists the lengths of stay for women who delivered babies by Cae-
sarean section.

Patient Length of Stay (LOS)
Kraut, Helene 2
Smith, Belinda 3
Serafin, Natalia 2
Jones, Janice 4
Rothschild, Pauline 32
Total days 43
ALOS 8.6

Just taking the mean LOS of these five patients, we calculate an ALOS of 8.6 days per
patient. But is 8.6 really representative of the average patient’s stay after a C-section? Cer-
tainly not; none of the other patients in this data set even stayed more than 4 days. Here is
another example: Dr. Garcia performs a variety of general surgeries, but his highest volume
is the cholecystectomy, the surgical removal of the gallbladder. Here is a set of observations
regarding Dr. Garcia’s volume:

January 12 cases
February 12 cases
March 12 cases
April 13 cases
May 12 cases
June 2 cases
Total 63 cases

Now consider the question: what is the average number of cases per month by Dr. Garcia in
the first half of 2016? If we use the arithmetic mean, the answer is 10.5 (63 cases divided
by 6 months). Does that make sense? Of course not. Dr. Garcia usually performs between
12 and 13 procedures. He has not performed less than 12 procedures, until June. For-
tunately, we can use one of our other M’s to get a better idea of how long these patients
usually stay.

Median

The second measure of central tendency is called the median. The median is the number
that represents the middle of an ordered array of the data you are examining. Another

p0475

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s0085

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Calculate the mean easily in Excel using the average formula: = average(cell range).
b0205

Stat Tip

Median The middle value of an
ordered array of data.b0220

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2

10002-DAVIS-9781455753154

way to state the definition is to say that 50% of the values are above the median, and 50%
of the values are below it. A median is useful because, unlike the mean, it is not affected
by extreme values. However, like the mean, there is only one. To determine the median
value, you must place the values in numerical order from lowest to highest (or highest to
lowest).

You might have noticed that the median instead of mean salaries are often reported
because of the influence of very low or very high examples. For example, the Quick Facts
about health careers on the Bureau of Labor Statistics Occupational Outlook Handbook
site (http://www.bls.gov/ooh/Management/Medical-and-health-services-managers.htm)
includes the median pay for each career.

Let us take the previous grades (90, 0, 90, 80, and 100). When we put them in order, you
can easily see that the middle grade is a 90. At this point, you can see that the median (90)
would be the same regardless of whether the lowest grade was an 80 or a zero. (Although
interesting to note, it is probably not a negotiating tool to get a better grade!)

What if you had an even set of numbers? Half of your samples will be divisible by two,
so consequently, the middle number will not be in the sample. An example of this would be
six grades instead of five. Look at 100, 90, 90, 80, 80, and 80. The two middle grades are 90
and 80. In order to calculate the median, add the two middle grades together and divide by
two. So, 90°+°80/2°=°85. In this case, the median is 85.

Let us go back again to Dr. Garcia’s surgeries. To determine the median for our example
data set, arrange the data in numerical order from lowest to highest:

June 2 cases
February 12 cases
January 12 cases
March 12 cases
May 12 cases
April 13 cases
Total 63 cases

There are 63 observations. The median is the midpoint in the list of observations: in this
case, observation #32. Counting down from the top of the list, the value associated with
observation #32 is 12. Therefore, the median number of cases for Dr. Garcia in the first
6 months of 2013 is 12. Note that there are an odd number of observations. If there were an
even number of observations, we would take the average of the two middle observations.
So, assuming there were 64 observations, we would average the value of observations 32
and 33. When we calculate the arithmetic mean and the results do not make sense based
on what we know the data otherwise reflects, we can use the median to give us more
insight into the distribution of the data. For further clarification we can use our third M,
the mode.

Mode

The last of the measures of central tendency is the mode. The mode is the most frequently
occurring observation in your sample. Using the 6 grades in the median example (100, 90,
90, 80, 80 and 80), you can see that you have one 100, two 90s, and three 80s. Because the 80
grades occur three times (more than two 90s or one 100), 80 is the mode for these grades.
Because the mode is simply the most frequently occurring, no calculation is needed. How-
ever, unlike the mean or median, there can be more than one mode. An instructor might
look at the class grades and see that he has 5 As, 8 Bs, 8 Cs, 1 D, and 1 F. In this case, the
modes would be B and C, because they both have the same highest number of grades. This
would be an example of a bimodal (bi- = two) distribution of grades. If there were three
highest values, it would be called trimodal (tri- = three). It is also possible that no mode
exists. So the mode is different from the mean and median in that they will always have one
and only one value, while the mode can have none, one, or more than one value. In a large
group of observations, a mode with many observations may indicate a strong preference or
tendency of the group. Because the mode is not a numerical calculation, it is possible that

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p0520 Mode The most frequently
occurring observation in a set
of data.

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http://www.bls.gov/ooh/Management/Medical-and-health-services-managers.htm

UNIT I Understanding the Basics of Statistics and Data Analytics

10002-DAVIS-9781455753154

TABLE 2-5

THE THREE M’S OF CENTRAL TENDENCY

CENTRAL
TENDENCY SYNONYMS?

HOW MANY
POSSIBLE?

AFFECTED BY
EXTREME
VALUES?

IS ORDER
NECESSARY TO
CALCULATE?

Mean Average One Yes No
Median None One No Yes
Mode None None, one,

more than one
No Yes

t0030

BRIEF CASE

FINDING MEAN, MEDIAN, AND MODE

Sasha is trying to determine the hospital equipment and staffing needs
for maternity care. Since newborn stays are largely determined by the
LOS of the mother, newborn statistics are often reviewed in conjunc-
tion with obstetrical delivery data. The table below shows the LOS for
newborns discharged over the course of a week. What are the mean,
median, and mode for this data set?

LOS: NEWBORNS, DISCHARGED 4/15–4/22

LOS NUMBER OF DISCHARGES TOTAL DAYS

1 day 3 3

2 days 7 14

3 days 7 21

4 days 2 8

5 days 1 5

6 days 0 0

7 days 1 7

21 58

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the group will have no mode (because all of the observations are at a single value). The lack
of a mode is not inherently important.

The mode answers questions like what is the most common number of procedures per-
formed by Dr. Garcia each month? In our example above, the most common number of
procedures performed is 12—the same number as the median. In this case, the median and
the mode are the same—casting further suspicion on the usefulness of the arithmetic mean
in this group of data. In this example, the median and the mode are better descriptions of
Dr. Garcia’s volume than the arithmetic mean.

However, the arithmetic mean does alert us to an anomaly in Dr. Garcia’s volume. In
reviewing the data, we can see the sharp drop in volume that occurred in July. Administra-
tors may be concerned that Dr. Garcia has decided to perform his surgeries at another hos-
pital. A simple phone call to the medical staff office or the health information management
department may reveal that Dr. Garcia is on vacation for a month and will resume surgeries
in August. A confirmation call to the scheduling department may yield the information that
Dr. Garcia is already fully booked for the first two weeks in August.

Although this example is certainly simple, changes from month to month in statistical
indicators such as the case-mix index (CMI) or average volume should trigger investiga-
tions into the reason for the change. Thus, statistics can be extremely helpful in monitoring
activities and highlighting changes before they become problematic.

Table 2-5 compares the three M’s of central tendency.

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Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2

10002-DAVIS-9781455753154

Adjusted Mean

Another way to look at the central tendency of data in which the mean, median, and mode
do not agree is to adjust the mean.

To adjust the mean of a set of data, we remove some of the data: the outliers. Typically,
we remove not only the outliers on one end, but also the corresponding number (or per-
centage) of observations on the other end: highest and lowest or largest and smallest. For
example, removing the first two and last two observations in Dr. Garcia’s surgery list gives
us 59 observations over 5 months (note that June is now eliminated). Thus, the adjusted
mean is 11.8: much closer to the 12 per month that we were expecting.

The purpose of adjusting the mean is only to get a sense of how unusual the outliers
really are. Up to 5% of the highest and 5% of the lowest is generally acceptable. In the
absence of policies or conventions, it is up to the presenter (the analyzer of the data) to
determine what percentage should be adjusted. However, a clear explanation of the adjust-
ment must accompany the report. The take away for all of the coverage of central tendency
is that these measures are seeking a way to describe the similarities in your group of data.
Each of the measures offers a different number (or different numbers) to give you a snap-
shot of a characteristic that gives a quick idea of what your group looks like.

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1. A home health nurse visited three patients on Monday, four on Tuesday, two on
Wednesday, and four on Thursday. What is the average number of patients he saw
on those four days? Provide your answer to the hundredths decimal place.

2. Over 12 months, an acute care facility compiled a report of the number of patients
transferred by month to a neighboring skilled nursing facility (SNF): 3, 10, 10, 11,
6, 10, 12, 11, 15, 8, 9, and 6.

a. What is the mean number of patients transferred per month?
b. What is the median?
c. What is the mode?
d. Which outliers would you remove to calculate the adjusted mean? What is your

calculation?

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EXERCISE 2-6
Measures of Central Tendency.

DISPERSION

The last basic math concept that needs to be addressed is that of dispersion, or the spread of
the data. Are all of your values close together or are they spread apart from each other? Dis-
persion deals with differences, not similarities. For example, a student with grades that are
82, 81, 79, 85, and 83 has grades that are fairly close together. Another student has grades of
82, 67, 98, 76, and 32—quite a bit of difference among those grades!

One of the simplest measures to describe dispersion is called the range. The range is the
difference between the lowest and highest (or highest and lowest) observation. The statisti-
cal range for the first student is 85–79 (or 79–85) with a range of six points. The second stu-
dent’s grades range from a high of 98 to a low of 32. That student’s range is 98–32 = 66. The
range is a simple, but crude method of looking at how different the scores are. You can see
it in three formats: highest to lowest, lowest to highest, or the difference between the two.

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It may be useful to provide the report both with and without the adjustment so that the
user can see exactly what impact the adjustment had on the reported data.b0235

Stat Tip

Dispersion The spread of the data. b0245

Range The difference between the
lowest and highest (or highest
and lowest) observation.

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UNIT I Understanding the Basics of Statistics and Data Analytics40

10002-DAVIS-9781455753154

Interquartile Range

If a simple range is used, extreme values can sway a truer measure of spread. Interquartile
and semi-interquartile ranges are used in healthcare when extreme values (outliers) are
present and the data analyst wants a less influenced picture of the data.

Fractiles are a means of dividing the data into fractional percentages. A decile is a type of
fractile dividing the data into percentages of 10, while quartiles divide the data into percent-
ages of 25 (quarters). Most commonly, a measure of spread that is used with the median is
the semi-interquartile range. The interquartile range and semi-interquartile range are two
measures that use the median, take out the influence of extreme values, and help provide a
cleaner picture of your data.

Using the data below, we can divide our 20 observations of patient lengths of stay into
quartiles. Each will have values that have 25% of the values. We can then observe that any
factor of 25 (i.e., 25, 50, 75) is above or below a particular value (Figure 2-4).

Patient

LOS

A. Booker 8

B. McCall 3

C. Rossman 46

D. Elias 6

E. Roman 1

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E. Roman 1

T. Weiner 1

L. Moore 2

U. Yellen 2

B. McCall 3

F. Shumacher 3

I. Edwards 3

P. Quigley 3

J. Frank 4

S. Underwood 4

N. Orville 5

K. Goode 6

D. Elias 6

O. Pau 7

A. Booker 8

M. Nunez 10

R. Tamaka 12

G. Ashton

H. Dorrance

C. Rossman 46

Patient Length of stay

Step 1: Order the observed values from lowest to highest

Step 2: Divide the ordered values into 4 groups

Step 3: Starting with the 50% value (the median), you can
see that half of your values are above and below this
number. In this array, the 10th and 11th values are 4 and
5, so the median is 4�5�2 � 4.5. The 25% quartile is
determined by observing the 5th and 6th values 3 and 3,
so the first quartile is 3 (25% of the values are 3 or less).
The 75% quartile is 8 and 10, so the 75% quartile is
8�10�2 � 9. 75% of the values are less than 9.

Figure 2-4 Finding a fractile: quartiles.
f0025

Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2

10002-DAVIS-9781455753154

Patient LOS
F. Shumacher 3
G. Ashton 17
H. Dorrance 19
I. Edwards 3
J. Frank 4
K. Goode 1
L. Moore 2
M. Nunez 10
N. Orville 5
O. Pau 7
P. Quigley 3
R. Tamaka 12
S. Underwood 4
T. Weiner 1
U. Yellen 2

The interquartile range is a measure of variation that is the absolute value of the difference
between the first and third quartiles. In this example, the interquartile range is 9–3, or
6 days. If the interquartile range is divided in half, it gives a statistic that gives an approxi-
mation of how far the scores spread from the median. For the example used, this would be
6/2 = +/− 3.

Variance

While central tendency looks at what the values have in common, another type of statistic,
the variance, looks at their differences. Variance is a measure of how different the values
are from each other. A simple measure of variance is used in budgeting when managers
compare their projected allotments to what was actually spent. In this use, variances may be
favorable or unfavorable. An over-spending is obviously unfavorable, while staying under
budget is favorable. This concept could also extend to increases/decreases in expected
or target values for admissions or deaths. An increase in admissions from one period to
another is likely favorable, while an increase in the number of deaths is probably unfavor-
able. (Thus, favorability is somewhat subjective.)

But variance is also a term that is used to describe another important statistical concept:
the difference between the calculated mean of a group of data and each individual observa-
tion. What the variance helps us understand here is how different is each item/patient from
the average for the group as a whole? To calculate this variance, we take the average of the
squared differences from the mean.

Let us look at a couple of examples to get a feel for this type of variance:
Emily’s grades are fairly close together: 82, 81, 79, 85, and 83.
Amanda’s grades have a wider dispersion: 82, 67, 98, 76, and 32.

Score Score Minus Mean Difference Squared
79 79 − 82 −3 9
81 81 − 82 −1 1
82 82 − 82 0 0
83 83 − 82 1 1
85 85 − 82 3 9

Sum 20

To calculate variance, we need to first obtain the mean. For Emily, we calculate 82 + 81 + 7
9 + 85 + 83 = 410 ÷ 5 = 82. Emily’s grade average is 82. Next, we need to determine what the
difference is between each score and the mean for all of the scores. Then, we square each
sum and add them all together.

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UNIT I Understanding the Basics of Statistics and Data Analytics42

10002-DAVIS-9781455753154

MATH REVIEW

Note that we rounded this number
down.

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This sum is 20, so the last step is to divide the sum by 5, and 20/5 = 4. The variance for
this particular student’s scores is 4. We will give additional meaning to this number in a
moment, but for now, the 4 represents the average difference between Emily’s scores and
the mean.

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You might ask why we squared the differences from the mean. Take a look at the
sum of the differences above: it is zero! The only way to get an amount that we can
work with is to square each result (do not worry, we will “unsquare” it later in our
calculations).

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Stat Tip

Let us take a look at Amanda’s grades with the huge range in her scores, and we will go
ahead and calculate a variance.

32+ 67+ 76+ 82+ 98= 355.355÷ 5= 71 for her mean (average)

Score Score Minus Mean Difference Squared
32 32−71 −39 1521
67 67−71 −4 16
76 76−71 5 25
82 82−71 11 121
98 98−71 27 729

Sum 2412

Add the sum of the squares: 1521 + 16 + 25 + 121 + 729 = 2412. Dividing 2412 by 5 = 482.4.
The variance for the first sample is 4, while the second is 482. This is a numerical measure
of just how different these two students are in the consistency of performing the same on
the tests that they have taken.

There are two important points:
1. If all of the scores are the same, the variance would be zero! And that would mean that

the scores are not different from each other at all.
2. You should know that variance is seldom used by itself, but it is most often used as a

means to calculate our final statistic, standard deviation. So let us move on to this often
misunderstood, but important statistic.

Standard Deviation

While range and variance give rough ideas of the differences in the high and low values in
your data, there is another statistic that gives another perspective and even more informa-
tion about your sample. Standard deviation is the square root of the variance and is repre-
sented by the Greek letter sigma, σ. Standard deviation is a measure of how spread out our
numbers are. It tells you if they are clumped together (a small standard deviation) or spread
far apart (a large standard deviation).

To continue using our examples with student test scores, Emily’s standard deviation
would be the square root of 5, or 2.23, rounded off to ± 2 (note that square roots can be
positive or negative). The second student’s standard deviation would be the square root of
603, or 24.55, rounded off to ± 25. The higher the standard deviation, the more varied the
data tends to be from the average.

The concept of standard deviation is based on a random sample, and it is used to predict
the probability of future events with a specified degree of confidence. You may have heard
of the bell curve (Figure 2-5), which is the classic representation of a normal distribution.
A normal distribution uses standard deviation to show how scores are expected to cluster
around the calculated average of a sample or population.

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Standard Deviation A measure of
variance showing how different
the observations are from the
mean.

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Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 43

10002-DAVIS-9781455753154

Again, using our first example, there is a 68% probability that Emily’s scores would be
within ± 2 of her average (82 + 2 = 84, and 82 − 2 = 80). So the scores of 80 to 84 are within 1
standard deviation from the mean. Two standard deviations from the mean could be calcu-
lated by adding and subtracting 2 more from each of these results (78 − 86), which gives us
a 96% probability of including all the scores.

Amanda has been much less consistent, and therefore, her results are much more dif-
ficult to predict. She had a standard deviation of ± 25. One standard deviation for Amanda
would be ± 25 from the mean of 71. That would give 46–96, and 2 standard deviations
would be 21–122! Notice in Figure 2-6 that the larger the standard deviation, the flatter the
bell curve, while the smaller it is, the more peaked it appears.

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F
re

qu
en

Standard deviations

�4 4�3 3�2 2�1 10

68% Probability

96% Probability

99.7% Probability

100% Probability

Figure 2-5 A bell curve.
f0030

Standard deviation is an
extremely useful tool in
helping to determine what
the expected values of
future values can be. Infer-
ential statistics are used to
determine this and include a
concept named “confidence
levels” that give a probabil-
ity of just how sure you can
be of your predictions.

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TAKE AWAY l

�

�4�5 4 5�3 3�2 2�1 10

1.0

0.8

0.6

0.4

0.2

0.0

�
, �

2
(�

)

��0,

�2�0.2,

�2�1.0,

�2�5.0,

Figure 2-6 Standard deviations and predictability.
f0035

UNIT I Understanding the Basics of Statistics and Data Analytics44

10002-DAVIS-9781455753154

LOS Length of stay.b0280

ALOS Average length of stay.b0285 [AU3]

Perhaps you are not relishing the task of calculating standard deviation by hand. Fortu-
nately, it can be accomplished fairly easily using Excel.
1. Click a cell directly below the column of numbers that you want a standard devia-

tion for and type the formula = STDEV([cell range]).
2. Highlight that entire column of values that you wish to examine and press enter.

Your calculated standard deviation will immediately appear. (You could also type
each value in individually; each number must be separated from the next with a
comma, and you will need to end with a close parenthesis.)

b0275

Stat Tip

Whether we are working with clinical, financial, or administrative data, there are numer-
ous instances when we might want to find the variance of an observation from the mean.
For example, we might want to set up a frequency distribution for a sample of patients’
lengths of stay (Figure 2-7).

In this set, we have a mean LOS of about 4 days. Now let us imagine we have a physician
whose patient’s ALOS is 7, varying 3 days longer than the average. If the standard deviation
of ALOS for these patients is 1.66, then an ALOS of 7 is nearly 2 standard deviations from
the mean (4 + 1.66 σ + 1.66 σ = 7.32). That means this physician’s ALOS is higher than 96%
of all others.

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Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 45

10002-DAVIS-9781455753154

20
21

32
33

42
43

A B C D F G H I J KE

LOS

DATA:

Length of stay of 250 patients discharged in April, 2012

Mean = Total of all LOS / Number of patients

=SUM(A7:J31)/250

=AVERAGE(A7:J31)

1 2 3 3 3 4 4 5 5 6

1 2 3 3 4 4 4 5 5 6

1 3 3 3 4 4 4 5 5 6

1 3 3 4 4 4 4 5 5 6

2 3 3 4 4 4 4 5 5 6

2 3 3 4 4 4 5 5 5 6

2 3 3 4 4 4 5 5 5 7

2 3 3 4 4 4 5 5 6 7

2 3 3 4 4 4 5 5 6 8

2 3 3 4 4 4 5 5 6 10

2 3 3 4 4 4 5 5 6 12

2 3 3 4 4 4 5 5 6 15

Mean = 4.144

=STDEVP(A7:J31)

Standard Deviation = 1.66

Frequency % of total patients

1 9 3.6%

2 20 8.0%

3 54 21.6%

4 77 30.8%

5 56 22.4%

6 21 8.4%

7 6 2.4%

8 4 1.6%

9 0 0.0%

10 1 0.4%

12 1 0.4%

15 1 0.4%

Total Patients 250

Series 1

1 2 3 4 5 6 7 8 9 10 11 12

Figure 2-7 A frequency distribution for LOS.
f0040

UNIT I Understanding the Basics of Statistics and Data Analytics46

10002-DAVIS-9781455753154

1. Convert the following improper fractions to mixed number fractions:

a. 18
8

b. 21
2

c. 14
12

2. Reduce the fractions below to their simplest form.

a. 33
66

b. 5
100

c.
750
1000

3. Add or subtract the following fractions. Report your answers in simple fractions.

a. 1
12

+
7
12

b. 1
8

−
2
3

c. 4
1
2

− 3
1
5

4. Convert the following fractions to decimals.

a.
1
8

b.
1
22

c. 14
12

5. Round each decimal to the tenths place. Then round each to the hundredths place.
a. 0.09513999
b. 0.551001
c. 22.7399
d. 1.1733
6. Convert each percentage to decimal form, or vice versa.
a. 55%
b. 17%
c. 1156%
d. 0.034
e. 0.78
f. 1.11
7. The college has a 97% placement rate for its new graduates. If there are 111 students in

this year’s class, how many will find jobs?
8. Convert 25 mg to grams.
9. A baby weighs 8 lbs. 1 oz at birth. How many grams does she weigh?

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Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 47

10002-DAVIS-9781455753154

10. To mix a certain medicine, you add one capful to ever 8 ounces of water. How many
capfuls do you need to make 64 ounces?

11. Your department has budgeted for three sets of references for every five coders. Re-
cently, a reorganization moved all of the coding to your hospital, and now there are 25
coders. How many sets of references will you need?

12. What does calculating the mean tell you? What are its advantages and disadvantages?
13. The nursing home has 50 male patients and 86 female patients. What is the absolute

frequency at the nursing home? What is the relative frequency of males? Of females?
What is the ratio of men to women in its simplest form?

14. While looking at salaries of physician’s assistants (PAs) in the health system, you find a
range from a low of $85,000 to a high of $160,000. How many classes would you have if
you broke them into class widths of $5000? What about widths of $20,000?

15. Fifty patients were treated at the free clinic last week. Their ages are listed from young-
est to oldest: 14, 15, 15, 16, 16, 17, 17, 17, 17, 18, 18, 18, 19, 19, 19, 19, 19, 19, 20, 20, 20,
21, 21, 21, 22, 22, 24, 24, 24, 25, 26, 26, 26, 27, 28, 30, 31, 32, 36, 40, 41, 41, 47, 52, 52,
59, 59, 60, 65, 72.

a. What is the mean age?
b. What is the median?
c. What is the mode?
d. What is the range?
e. Calculate the variance of the ages listed from a frequency distribution.
f. Calculate the variance of the ages from a frequency distribution with class limits of

14 to 35, and a standard deviation with a class interval of 1.

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2 – Basic Math Concepts, Central Tendency, and Dispersion

Fractions, Decimals, and Percentages

Fractions

Decimals

Changing Fractions to Decimals

Percentages

Ratio, Rate, and Proportion

Ratio

Rate

Proportion

Volume, Frequency, and Frequency Distribution

Volume

Frequency

Frequency Distribution

Measures of Central Tendency

Mean

Median

Mode

Adjusted Mean

Dispersion

Interquartile Range

Variance

Standard Deviation

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