Contents

Measures of the Location of the Data

The common measures of location are quartiles and percentiles.

Quartiles are special percentiles. The first quartile, Q₁, is the same as the 25^th percentile, and the third quartile, Q₃, is the same as the 75^th percentile. The median, M, is called both the second quartile and the 50^th percentile.

To calculate quartiles and percentiles, you must order the data from smallest to largest. Quartiles divide ordered data into quarters. Percentiles divide ordered data into hundredths. Recall that a percent means one-hundredth. So, percentiles mean the data is divided into 100 sections. To score in the 90^th percentile of an exam does not mean, necessarily, that you received 90 percent on a test. It means that 90 percent of test scores are the same as or less than your score and that 10 percent of the test scores are the same as or greater than your test score.

Percentiles are useful for comparing values. For this reason, universities and colleges use percentiles extensively. One instance in which colleges and universities use percentiles is when SAT results are used to determine a minimum testing score that will be used as an acceptance factor. For example, suppose Duke accepts SAT scores at or above the 75^th percentile. That translates into a score of at least 1220.

Percentiles are mostly used with very large populations. Therefore, if you were to say that 90 percent of the test scores are less, and not the same or less, than your score, it would be acceptable because removing one particular data value is not significant.

The median is a number that measures the center of the data. You can think of the median as the middle value, but it does not actually have to be one of the observed values. It is a number that separates ordered data into halves. Half the values are the same number or smaller than the median, and half the values are the same number or larger. For example, consider the following data:
1, 11.5, 6, 7.2, 4, 8, 9, 10, 6.8, 8.3, 2, 2, 10, 1
Ordered from smallest to largest:
1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5

When a data set has an even number of data values, the median is equal to the average of the two middle values when the data are arranged in ascending order (least to greatest). When a data set has an odd number of data values, the median is equal to the middle value when the data are arranged in ascending order.

Since there are 14 observations (an even number of data values), the median is between the seventh value, 6.8, and the eighth value, 7.2. To find the median, add the two values together and divide by two.

\frac{6.8 + 7.2}{2} = 7 6.8
7.2

2

7

2.1

The median is seven. Half of the values are smaller than seven and half of the values are larger than seven.

Quartiles are numbers that separate the data into quarters. Quartiles may or may not be part of the data. To find the quartiles, first find the median, or second, quartile. The first quartile, Q₁, is the middle value of the lower half of the data, and the third quartile, Q₃, is the middle value, or median, of the upper half of the data. To get the idea, consider the same data set:
1, 1, 2, 2, 4, 6, 6.8, 7.2, 8, 8.3, 9, 10, 10, 11.5

The data set has an even number of values (14 data values), so the median will be the average of the two middle values (the average of 6.8 and 7.2), which is calculated as $\frac{6.8 + 7.2}{2} \frac{6.8 + 7.2}{2}$ and equals 7.

So, the median, or second quartile ( $Q_{2} Q_{2}$ ), is 7.

The first quartile is the median of the lower half of the data, so if we divide the data into seven values in the lower half and seven values in the upper half, we can see that we have an odd number of values in the lower half. Thus, the median of the lower half, or the first quartile ( $Q_{1} Q_{1}$ ) will be the middle value, or 2. Using the same procedure, we can see that the median of the upper half, or the third quartile ( $Q_{3} Q_{3}$ ) will be the middle value of the upper half, or 9.

The quartiles are illustrated below:

A number line is shown including the numbers 1, 1, 2, 2, 4, 6, 6.8, 7.23, 8, 8.3, 9, 10, 10, and 11.5. The following numbers are circled in red: 2, 6.8, 7.2 and 9. There is a line between 6.8 and 7.2. Q1 is located above the second number 2. Q 3 is located above the number 9. The equation Q 2 is equal to 6.8 plus 7.2 over 2 is located above the middle of the number line.

Figure 2.11

The interquartile range is a number that indicates the spread of the middle half, or the middle 50 percent of the data. It is the difference between the third quartile (Q₃) and the first quartile (Q₁)

IQR = Q₃ – Q₁. The IQR for this data set is calculated as 9 minus 2, or 7.

The IQR can help to determine potential outliers. A value is suspected to be a potential outlier if it is less than 1.5 × IQR below the first quartile or more than 1.5 × IQR above the third quartile. Potential outliers always require further investigation.

NOTE

A potential outlier is a data point that is significantly different from the other data points. These special data points may be errors or some kind of abnormality, or they may be a key to understanding the data.

Example 2.15

Problem

For the following 13 real estate prices, calculate the IQR and determine if any prices are potential outliers. Prices are in dollars.
389,950; 230,500; 158,000; 479,000; 639,000; 114,950; 5,500,000; 387,000; 659,000; 529,000; 575,000; 488,800; 1,095,000

Show solution

Solution

Order the following data from smallest to largest:
114,950; 158,000; 230,500; 387,000; 389,950; 479,000; 488,800; 529,000; 575,000; 639,000; 659,000; 1,095,000; 5,500,000

M = 488,800

Q₁ = $\frac{230,500 + 387,000}{2} \frac{230,500 + 387,000}{2}$ = 308,750

Q₃ = $\frac{639,000 + 659,000}{2} \frac{639,000 + 659,000}{2}$ = 649,000

IQR = 649,000 – 308,750 = 340,250

(1.5)(IQR) = (1.5)(340,250) = 510,375

Q₁ – (1.5)(IQR) = 308,750 – 510,375 = –201,625

Q₃ + (1.5)(IQR) = 649,000 + 510,375 = 1,159,375

No house price is less than –201,625. However, 5,500,000 is more than 1,159,375. Therefore, 5,500,000 is a potential outlier.

Try It 2.15

For the 11 salaries, calculate the IQR and determine if any salaries are outliers. The following salaries are in dollars.

$33,000; $64,500; $28,000; $54,000; $72,000; $68,500; $69,000; $42,000; $54,000; $120,000; $40,500

In the example above, you just saw the calculation of the median, first quartile, and third quartile. These three values are part of the five number summary. The other two values are the minimum value (or min) and the maximum value (or max). The five number summary is used to create a box plot.

Try It 2.15

Find the interquartile range for the following two data sets and compare them.

Test Scores for Class A:
69, 96, 81, 79, 65, 76, 83, 99, 89, 67, 90, 77, 85, 98, 66, 91, 77, 69, 80, 94
Test Scores for Class B:
90, 72, 80, 92, 90, 97, 92, 75, 79, 68, 70, 80, 99, 95, 78, 73, 71, 68, 95, 100

Example 2.16

Fifty statistics students were asked how much sleep they get per school night (rounded to the nearest hour). The results were as follows:

Amount of Sleep per School Night (Hours)	Frequency	Relative Frequency	Cumulative Relative Frequency
4	2	.04	.04
5	5	.10	.14
6	7	.14	.28
7	12	.24	.52
8	14	.28	.80
9	7	.14	.94
10	3	.06	1.00

Table 2.24

Find the 28^th percentile. Notice the .28 in the Cumulative Relative Frequency column. Twenty-eight percent of 50 data values is 14 values. There are 14 values less than the 28^th percentile. They include the two 4s, the five 5s, and the seven 6s. The 28^th percentile is between the last six and the first seven. The 28^th percentile is 6.5.

Find the median. Look again at the Cumulative Relative Frequency column and find .52. The median is the 50^th percentile or the second quartile. Fifty percent of 50 is 25. There are 25 values less than the median. They include the two 4s, the five 5s, the seven 6s, and 11 of the 7s. The median or 50^th percentile is between the 25^th, or seven, and 26^th, or seven, values. The median is seven.

Find the third quartile. The third quartile is the same as the 75^th percentile. You can eyeball this answer. If you look at the Cumulative Relative Frequency column, you find .52 and .80. When you have all the fours, fives, sixes, and sevens, you have 52 percent of the data. When you include all the 8s, you have 80 percent of the data. The 75^th percentile, then, must be an eight. Another way to look at the problem is to find 75 percent of 50, which is 37.5, and round up to 38. The third quartile, Q₃, is the 38^th value, which is an eight. You can check this answer by counting the values. There are 37 values below the third quartile and 12 values above.

Try It 2.16

Forty bus drivers were asked how many hours they spend each day running their routes (rounded to the nearest hour). Find the 65^th percentile.

Amount of Time Spent on Route (Hours)	Frequency	Relative Frequency	Cumulative Relative Frequency
2	12	.30	.30
3	14	.35	.65
4	10	.25	.90
5	4	.10	1.00

Table 2.25

Example 2.17

Problem

Using Table 2.24:

Find the 80^th percentile.
Find the 90^th percentile.
Find the first quartile. What is another name for the first quartile?

Show solution

Solution

Using the data from the frequency table, we have the following:

The 80^th percentile is between the last eight and the first nine in the table (between the 40^th and 41^st values). Therefore, we need to take the mean of the 40^th an 41^st values. The 80^th percentile $= \frac{8 + 9}{2} = 8.5 . = \frac{8 + 9}{2} = 8.5 .$
The 90^th percentile will be the 45^th data value (location is 0.90(50) = 45), and the 45^th data value is nine.
Q₁ is also the 25^th percentile. The 25^th percentile location calculation: P₂₅ = .25(50) = 12.5 ≈ 13, the 13^th data value. Thus, the 25^th percentile is six.

Try It 2.17

Refer to Table 2.25. Find the third quartile. What is another name for the third quartile?

Collaborative Exercise

Your instructor or a member of the class will ask everyone in class how many sweaters he or she owns. Answer the following questions:

How many students were surveyed?
What kind of sampling did you do?
Construct two different histograms. For each, starting value = ________ and ending value = ________.
Find the median, first quartile, and third quartile.
Construct a table of the data to find the following:
1. The 10^th percentile
2. The 70^th percentile
3. The percentage of students who own fewer than four sweaters

A Formula for Finding the kth Percentile

If you were to do a little research, you would find several formulas for calculating the k^th percentile. Here is one of them.

k = the k^th percentile. It may or may not be part of the data.

i = the index (ranking or position of a data value)

n = the total number of data

Order the data from smallest to largest.
Calculate $i = \frac{k}{100} (n + 1) . i = \frac{k}{100} (n + 1) .$
If i is an integer, then the k^th percentile is the data value in the i^th position in the ordered set of data.
If i is not an integer, then round i up and round i down to the nearest integers. Average the two data values in these two positions in the ordered data set. The formula and calculation are easier to understand in an example.

Example 2.18

Problem

Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:
18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77

Find the 70^th percentile.
Find the 83^rd percentile.

Show solution

Solution

- k = 70
- i = the index
- n = 29
i = k 100 k 100 (n + 1) = ( 70 100 70 100 )(29 + 1) = 21. This equation tells us that i, or the position of the data value in the data set, is 21. So, we will count over to the 21^st position, which shows a data value of 64.
- k = 83^rd percentile
- i = the index
- n = 29
i = k 100 k 100 (n + 1) = ( 83 100 83 100 )(29 + 1) = 24.9, which is not an integer. Round it down to 24 and up to 25. The age in the 24^th position is 71, and the age in the 25^th position is 72. Average 71 and 72. The 83^rd percentile is 71.5 years.

Try It 2.18

Listed are 29 ages for Academy Award-winning best actors in order from smallest to largest:

18, 21, 22, 25, 26, 27, 29, 30, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77
Calculate the 20^th percentile and the 55^th percentile.

NOTE

You can calculate percentiles using calculators and computers. There are a variety of online calculators.

A Formula for Finding the Percentile of a Value in a Data Set

Order the data from smallest to largest.
x = the number of data values counting from the bottom of the data list up to but not including the data value for which you want to find the percentile.
y = the number of data values equal to the data value for which you want to find the percentile.
n = the total number of data.
Calculate $\frac{x + .5 y}{n} \frac{x + .5 y}{n}$ (100). Then round to the nearest integer.

Example 2.19

Problem

Find the percentile for 58.
Find the percentile for 25.

Show solution

Solution

Counting from the bottom of the list, there are 18 data values less than 58. There is one value of 58.
x = 18 and y = 1. $\frac{x + .5 y}{n} \frac{x + .5 y}{n}$ (100) = $\frac{18 + .5 (1)}{29} \frac{18 + .5 (1)}{29}$ (100) = 63.80. Fifty-eight is the 64^th percentile.
Counting from the bottom of the list, there are three data values less than 25. There is one value of 25.
x = 3 and y = 1. $\frac{x + .5 y}{n} \frac{x + .5 y}{n}$ (100) = $\frac{3 + .5 (1)}{29} \frac{3 + .5 (1)}{29}$ (100) = 12.07. Twenty-five is the 12^th percentile.

Try It 2.19

Listed are 30 ages for Academy Award-winning best actors in order from smallest to largest:

18, 21, 22, 25, 26, 27, 29, 30, 31, 31, 33, 36, 37, 41, 42, 47, 52, 55, 57, 58, 62, 64, 67, 69, 71, 72, 73, 74, 76, 77
Find the percentiles for 47 and 31.

Interpreting Percentiles, Quartiles, and Median

A percentile indicates the relative standing of a data value when data are sorted into numerical order from smallest to largest. Percentages of data values are less than or equal to the pth percentile. For example, 15 percent of data values are less than or equal to the 15^th percentile.

Low percentiles always correspond to lower data values.
High percentiles always correspond to higher data values.

A percentile may or may not correspond to a value judgment about whether it is good or bad. The interpretation of whether a certain percentile is good or bad depends on the context of the situation to which the data apply. In some situations, a low percentile would be considered good; in other contexts a high percentile might be considered good. In many situations, there is no value judgment that applies. A high percentile on a standardized test is considered good, while a lower percentile on body mass index might be considered good. A percentile associated with a person's height doesn't carry any value judgment.

Understanding how to interpret percentiles properly is important not only when describing data, but also when calculating probabilities in later chapters of this text.

Guideline

When writing the interpretation of a percentile in the context of the given data, make sure the sentence contains the following information:

Information about the context of the situation being considered
The data value (value of the variable) that represents the percentile
The percentage of individuals or items with data values below the percentile
The percentage of individuals or items with data values above the percentile

Example 2.20

Problem

On a timed math test, the first quartile for time it took to finish the exam was 35 minutes. Interpret the first quartile in the context of this situation.

Show solution

Solution

Twenty-five percent of students finished the exam in 35 minutes or less.
Seventy-five percent of students finished the exam in 35 minutes or more.
A low percentile could be considered good, as finishing more quickly on a timed exam is desirable. If you take too long, you might not be able to finish.

Try It 2.20

For the 100-meter dash, the third quartile for times for finishing the race was 11.5 seconds. Interpret the third quartile in the context of the situation.

Example 2.21

Problem

On a 20-question math test, the 70^th percentile for number of correct answers was 16. Interpret the 70^th percentile in the context of this situation.

Show solution

Solution

Seventy percent of students answered 16 or fewer questions correctly.
Thirty percent of students answered 16 or more questions correctly.
A higher percentile could be considered good, as answering more questions correctly is desirable.

Try It 2.21

On a 60-point written assignment, the 80^th percentile for the number of points earned was 49. Interpret the 80^th percentile in the context of this situation.

Example 2.22

Problem

At a high school, it was found that the 30^th percentile of number of hours that students spend studying per week is seven hours. Interpret the 30^th percentile in the context of this situation.

Show solution

Solution

Thirty percent of students study seven or fewer hours per week.
Seventy percent of students study seven or more hours per week.
In this example, there is not necessarily a good or bad value judgment associated with a higher or lower percentile, since the time a student studies per week is dependent on his/her needs.

Try It 2.22

During a season, the 40^th percentile for points scored per player in a game is eight. Interpret the 40^th percentile in the context of this situation.

Example 2.23

A middle school is applying for a grant that will be used to add fitness equipment to the gym. The principal surveyed 15 anonymous students to determine how many minutes a day the students spend exercising. The results from the 15 anonymous students are shown:

0 minutes, 40 minutes, 60 minutes, 30 minutes, 60 minutes,

10 minutes, 45 minutes, 30 minutes, 300 minutes, 90 minutes,

30 minutes, 120 minutes, 60 minutes, 0 minutes, 20 minutes

Find the five values that make up the five number summary.

Min = 0

Q₁ = 20

Med = 40

Q₃ = 60

Max = 300

Listing the data in ascending order gives the following:

A number line is shown including the numbers 0, 0, 10, 20, 30, 30, 30, 40, 45, 60, 60, 60, 90, 120, and 300. The following numbers are circled: 20, 40, and 60. The number 40 is encapsulated by 2 vertical lines.

Figure 2.12

The minimum value is 0.

The maximum value is 300.

Since there are an odd number of data values, the median is the middle value of this data set as it is arranged in ascending order, or 40.

The first quartile is the median of the lower half of the scores and does not include the median. The lower half has seven data values; the median of the lower half will equal the middle value of the lower half, or 20.

The third quartile is the median of the upper half of the scores and does not include the median. The upper half also has seven data values; so the median of the upper half will equal the middle value of the upper half, or 60.

If you were the principal, would you be justified in purchasing new fitness equipment? Since 75 percent of the students exercise for 60 minutes or less daily, and since the IQR is 40 minutes (60 – 20 = 40), we know that half of the students surveyed exercise between 20 minutes and 60 minutes daily. This seems a reasonable amount of time spent exercising, so the principal would be justified in purchasing the new equipment.

However, the principal needs to be careful. The value 300 appears to be a potential outlier.

Q₃ + 1.5(IQR) = 60 + (1.5)(40) = 120.

The value 300 is greater than 120, so it is a potential outlier. If we delete it and calculate the five values, we get the following values:

Min = 0
Q₁ = 20
Q₃ = 60
Max = 120

We still have 75 percent of the students exercising for 60 minutes or less daily and half of the students exercising between 20 and 60 minutes a day. However, 15 students is a small sample, and the principal should survey more students to be sure of his survey results.