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

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class cumulative frequency decimal dispersion fraction frequency frequency distribution mean

median mode percentage proportion quartiles quotient 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.

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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

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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.

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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.

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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.

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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,

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Basic Math Concepts, Central Tendency, and Dispersion CHAPTER 2 21

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3 2

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 56 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.

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

15 4

4 4

3 4

�

� 1 � 1 � 1 �

� � �( ) 4 4( ) 4

4( ) 3 4

3 4

Step 1: Multiply the denominator (4) by the whole number (3) 4 � 3 � 12

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 Analytics22

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.

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ICU Intensive care unit.b0040

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

16 64

÷ 8= 16÷ 8 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 12

= 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!)

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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

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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.

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

a. 12 8

b. 5 2

c. 144 12

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

a. 3 3 8

b. 13 1 2

c. 7 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

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EXERCISE 2-1 Fractions.

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

b0060 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:

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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:

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

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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

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Quotient The result of division. b0070

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

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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

dt hs

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.

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

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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.

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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:

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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

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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.

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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. 0.551031 c. 1.342809

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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?

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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.

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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

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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.

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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.

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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.

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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%

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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.

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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.

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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.

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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%

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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.

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

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ICU Intensive care unit. b0115 [AU2]

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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.

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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.

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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.

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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.

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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?

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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

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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.

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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.

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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

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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

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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).

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

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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

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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

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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).

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

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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,

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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?

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EXERCISE 2-5

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

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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.

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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

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Stat Tip

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

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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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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

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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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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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Dispersion The spread of the data. b0245

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

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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 17

H. Dorrance 19

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

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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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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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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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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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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.)

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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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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

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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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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