Measures of the Location of the Data, Descriptive Statistics | Statistics

Learning objectives

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.

6.8
7.2

2

7

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 $\frac{6.8 + 7.2}{2}$ and equals 7.

So, the median, or second quartile ( $Q_{2}$ ), 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 ( $Q_{1}$ ) 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 ( $Q_{3}$ ) will be the middle value of the upper half, or 9.

The quartiles are illustrated below:

View original figure (external; image not reproduced in this text edition)

Original figure description: 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.

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.

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

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

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

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

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

Class A

Order the data from smallest to largest:

65, 66, 67, 69, 69, 76, 77, 77, 79, 80, 81, 83, 85, 89, 90, 91, 94, 96, 98, 99

$Median = \frac{80 + 81}{2} = 80.5$

$Q_{1} = \frac{69 + 76}{2} = 72.5$

$Q_{3} = \frac{90 + 91}{2} = 90.5$

IQR = 90.5 – 72.5 = 18

Class B

Order the data from smallest to largest:

68, 68, 70, 71, 72, 73, 75, 78, 79, 80, 80, 90, 90, 92, 92, 95, 95, 97, 99, 100

$Median = \frac{80 + 80}{2} = 80$

$Q_{1} = \frac{72 + 73}{2} = 72.5$

$Q_{3} = \frac{92 + 95}{2} = 93.5$

IQR = 93.5 – 72.5 = 21

The data for Class B has a larger IQR, so the scores between Q₃ and Q₁ (middle 50%) for the data for Class B are more spread out and not clustered about the median.

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

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

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

The 65^th percentile is between the last three and the first four.

The 65^th percentile is 3.5.

Using id4431204:

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

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 $= \frac{8 + 9}{2} = 8.5 .$
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.

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 $i = \frac{k}{100} (n + 1) .$
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.

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.

- k = 70
- i = the index
- n = 29
i = k 100 (n + 1) = ( 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 (n + 1) = ( 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.

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 $\frac{x + .5 y}{n}$ (100). Then round to the nearest integer.

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

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. $\frac{x + .5 y}{n}$ (100) = $\frac{18 + .5 (1)}{29}$ (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. $\frac{x + .5 y}{n}$ (100) = $\frac{3 + .5 (1)}{29}$ (100) = 12.07. Twenty-five is the 12^th percentile.

Try It

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.

Percentile for 47: Counting from the bottom of the list, there are 15 data values less than 47. There is one value of 47.

x = 15 and y = 1. $\frac{x + 0.5 y}{n}$ (100) = $\frac{15 + 0.5 (1)}{29}$ (100) = 53.45. 47 is the 53^rd percentile.

Percentile for 31: Counting from the bottom of the list, there are eight data values less than 31. There are two values of 31.

x = 15 and y = 2. $\frac{x + 0.5 y}{n}$ (100) = $\frac{15 + 0.5 (2)}{29}$ (100) = 31.03. 31 is the 31^st percentile.

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.

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.

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.

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.

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.

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.

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.

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:

View original figure (external; image not reproduced in this text edition)

Original figure description: 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.

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.

References

Cauchon, D., and Overberg, P. (2012). Census data shows minorities now a majority of U.S. births. USA Today. Retrieved from http://usatoday30.usatoday.com/news/nation/story/2012-05-17/minority-birthscensus/55029100/1

The Mercury News. (n.d.). Retrieved from http://www.mercurynews.com/

Time. (n.d.). Survey by Yankelovich Partners, Inc.

U.S. Census Bureau. (1990). 1990 census. Retrieved from http://www.census.gov/main/www/cen1990.html

U.S. Census Bureau. (n.d.). Data. Retrieved from http://www.census.gov/

Chapter Review

The values that divide a rank-ordered set of data into 100 equal parts are called percentiles. Percentiles are used to compare and interpret data. For example, an observation at the 50^th percentile would be greater than 50 percent of the other observations in the set. Quartiles divide data into quarters. The first quartile (Q₁) is the 25^th percentile, the second quartile (Q₂ or median) is the 50^th percentile, and the third quartile (Q₃) is the 75^th percentile. The interquartile range, or IQR, is the range of the middle 50 percent of the data values. The IQR is found by subtracting Q₁ from Q₃ and can help determine outliers by using the following two expressions.

Q₃ + IQR(1.5)
Q₁ – IQR(1.5)

Formula Review

$i = (\frac{k}{100}) (n + 1)$

where i = the ranking or position of a data value,

k = the kth percentile,

n = total number of data.

Expression for finding the percentile of a data value $(\frac{x + 0.5 y}{n})$ (100)

where x = the number of 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 = total number of data.

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 40^th percentile.
Find the 78^th percentile.

The 40^th percentile is 37 years.
The 78^th percentile is 70 years.

Listed are 32 ages for Academy Award-winning best actors in order from smallest to largest:

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

Find the percentile of 37.
Find the percentile of 72.

Jesse was ranked 37^th in his graduating class of 180 students. At what percentile is Jesse’s ranking?

Jesse graduated 37^th out of a class of 180 students. There are 180 – 37 = 143 students ranked below Jesse. There is one rank of 37.

x = 143 and y = 1. $\frac{x + .5 y}{n}$ (100) = $\frac{143 + .5 (1)}{180}$ (100) = 79.72. Jesse’s rank of 37 puts him at the 80^th percentile.

For runners in a race, a low time means a faster run. The winners in a race have the shortest running times. Is it more desirable to have a finish time with a high or a low percentile when running a race?
The 20^th percentile of run times in a particular race is 5.2 minutes. Write a sentence interpreting the 20^th percentile in the context of the situation.
A bicyclist in the 90^th percentile of a bicycle race completed the race in 1 hour and 12 minutes. Is he among the fastest or slowest cyclists in the race? Write a sentence interpreting the 90^th percentile in the context of the situation.

For runners in a race, a higher speed means a faster run. Is it more desirable to have a speed with a high or a low percentile when running a race?
The 40^th percentile of speeds in a particular race is 7.5 miles per hour. Write a sentence interpreting the 40^th percentile in the context of the situation.

For runners in a race, it is more desirable to have a high percentile for speed. A high percentile means a higher speed, which is faster.
40 percent of runners ran at speeds of 7.5 miles per hour or less (slower), and 60 percent of runners ran at speeds of 7.5 miles per hour or more (faster).

On an exam, would it be more desirable to earn a grade with a high or a low percentile? Explain.

Mina is waiting in line at the Department of Motor Vehicles. Her wait time of 32 minutes is the 85^th percentile of wait times. Is that good or bad? Write a sentence interpreting the 85^th percentile in the context of this situation.

When waiting in line at the DMV, the 85^th percentile would be a long wait time compared to the other people waiting. 85 percent of people had shorter wait times than Mina. In this context, Mina would prefer a wait time corresponding to a lower percentile. 85 percent of people at the DMV waited 32 minutes or less. 15 percent of people at the DMV waited 32 minutes or longer.

In a survey collecting data about the salaries earned by recent college graduates, Li found that her salary was in the 78^th percentile. Should Li be pleased or upset by this result? Explain.

In a study collecting data about the repair costs of damage to automobiles in a certain type of crash tests, a certain model of car had $1,700 in damage and was in the 90^th percentile. Should the manufacturer and the consumer be pleased or upset by this result? Explain and write a sentence that interprets the 90^th percentile in the context of this problem.

The manufacturer and the consumer would be upset. This is a large repair cost for the damages, compared to the other cars in the sample. INTERPRETATION: 90 percent of the crash-tested cars had damage repair costs of $1,700 or less; only 10 percent had damage repair costs of $1,700 or more.

The University of California has two criteria used to set admission standards for freshman to be admitted to a college in the UC system:

Students' GPAs and scores on standardized tests (SATs and ACTs) are entered into a formula that calculates an admissions index score. The admissions index score is used to set eligibility standards intended to meet the goal of admitting the top 12 percent of high school students in the state. In this context, what percentile does the top 12 percent represent?
Students whose GPAs are at or above the 96^th percentile of all students at their high school are eligible, called eligible in the local context, even if they are not in the top 12 percent of all students in the state. What percentage of students from each high school are eligible in the local context?

Suppose that you are buying a house. You and your real estate agent have determined that the most expensive house you can afford is the 34^th percentile. The 34^th percentile of housing prices is $240,000 in the town you want to move to. In this town, can you afford 34 percent of the houses or 66 percent of the houses?

You can afford 34 percent of houses. 66 percent of the houses are too expensive for your budget. INTERPRETATION: 34 percent of houses cost $240,000 or less; 66 percent of houses cost $240,000 or more.

Use fs-idp30887728 to calculate the following values.

First quartile = ________

Second quartile = median = 50^th percentile = ________

Third quartile = ________

Interquartile range (IQR) = ________ – ________ = ________

6 – 4 = 2

10^th percentile = ________

70^th percentile = ________

Homework

The median age for U.S. ethnicity A currently is 30.9 years; for U.S. ethnicity B, it is 42.3 years.

Based on this information, give two reasons why ethnicity A median age could be lower than the ethnicity B median age.
Does the lower median age for ethnicity A necessarily mean that ethnicity A die younger than ethnicity B? Why or why not?
How might it be possible for ethnicity A and ethnicity B to die at approximately the same age but for the median age for ethnicity B to be higher?

Six hundred adult Americans were asked by telephone poll, "What do you think constitutes a middle-class income?" The results are in element-588. Also, include the left endpoint but not the right endpoint.

Salary ($)	Relative Frequency
< 20,000	.02
20,000–25,000	.09
25,000–30,000	.19
30,000–40,000	.26
40,000–50,000	.18
50,000–75,000	.17
75,000–99,999	.02
100,000+	.01

What percentage of the survey answered "not sure"?
What percentage think that middle class is from $25,000 to $50,000?
Construct a histogram of the data.
1. Should all bars have the same width, based on the data? Why or why not?
2. How should the < 20,000 and the 100,000+ intervals be handled? Why?
Find the 40^th and 80^th percentiles.
Construct a bar graph of the data.

1 – (.02+.09+.19+.26+.18+.17+.02+.01) = .06
.19+.26+.18 = .63
Check student’s solution.
40^th percentile will fall between 30,000 and 40,000

80^th percentile will fall between 50,000 and 75,000
Check student’s solution.

Given the following box plot, answer the questions.

View original figure (external; image not reproduced in this text edition)

Original figure description: This is a horizontal boxplot graphed over a number line from 0 to 13. The first whisker extends from the smallest value, 0, to the first quartile, 2. The box begins at the first quartile and extends to third quartile, 12. A vertical, dashed line is drawn at median, 10. The second whisker extends from the third quartile to largest value, 13.

Which quarter has the smallest spread of data? What is that spread?
Which quarter has the largest spread of data? What is that spread?
Find the interquartile range (IQR).
Are there more data in the interval 5–10 or in the interval 10–13? How do you know this?
Which interval has the fewest data in it? How do you know this?
1. 0–2
2. 2–4
3. 10–12
4. 12–13
5. need more information

The following box plot shows the ages of the U.S. population for 1990, the latest available year:

View original figure (external; image not reproduced in this text edition)

Original figure description: A box plot with values from 0 to 105, with Q1 at 17, M at 33, and Q3 at 50.

Are there fewer or more children (age 17 and under) than senior citizens (age 65 and over)? How do you know?
12.6 percent are age 65 and over. Approximately what percentage of the population are working-age adults (above age 17 to age 65)?

more children; the left whisker shows that 25 percent of the population are children 17 and younger; the right whisker shows that 25 percent of the population are adults 50 and older, so adults 65 and over represent less than 25 percent
62.4 percent

Glossary

interquartile range: or IQR, is the range of the middle 50 percent of the data values; the IQR is found by subtracting the first quartile from the third quartile
outlier: an observation that does not fit the rest of the data
percentile: a number that divides ordered data into hundredths; percentiles may or may not be part of the data. The median of the data is the second quartile and the 50^th percentile
The first and third quartiles are the 25^th and the 75^th percentiles, respectively.
quartiles: the numbers that separate the data into quarters; quartiles may or may not be part of the data; the second quartile is the median of the data

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