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Quartiles and
Box-and-Whisker Plots
NY-2
Learning Standards for
Mathematics
GO for Help
Check Skills You’ll Need
A.S.11 Find the percentile
rank of an item in a data
set and identify the point
values for ﬁrst, second,
and third quartiles.
Find the range and the median.
1. 3 8 12 15 20
3. 0 2 7 10 –1 –4 –11
A.S.5, A.S.9, A.S.6
Construct, analyze, and
interpret a box-andwhisker plot using the
ﬁve number summary.
Lesson 1-6
2. 2.1 3.3 –5.4 0.8 3.5
4. 64 16 23 57 14 22
New Vocabulary • ﬁrst quartile • third quartile • ﬁve number summary
• minimum • maximum • percentiles • percentile rank
• quartile • box-and-whisker plot • interquartile range
1
2
Part
Graphing and
y ≠ ax
Percentiles
1 1 Quartiles
In Lesson 1-6, you used measures of central tendency to describe data sets. In this
lesson, you will use quartiles to separate data in four equal parts. The median
separates the data into upper and lower halves. The ﬁrst quartile (Q1) is the
median of the lower half of the data. The third quartile (Q3) is the median of
the upper half of the data.
minimum
1
EXAMPLE
Q1
Q2
Q3
first quartile
median of
lower half
median
third quartile
median of
upper half
maximum
Finding Median, First, and Third Quartiles
For the data below, ﬁnd the median (Q2), ﬁrst quartile (Q1), and third quartile (Q3).
39 27.5 26 31 42 26 37 30 22
Step 1 Arrange the data in order from least to greatest. Find the median.
22 26 26 27.5 30 31 37 39 42
The median (Q2) is the middle value 30.
Step 2 Find the ﬁrst quartile and third quartile, which are the medians of the
lower and upper halves.
22 26
| 26 27.5 30 31 37 | 39 42
first quartile (Q1) = 26 1 26 = 26
2
Quick Check
NY 732
third quartile (Q3) = 37 1 39 = 38
2
1 For the following data, ﬁnd the median, the ﬁrst quartile, and the third quartile.
480 500 600 250 660 570 490 610 530
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The ﬁve number summary of a set of data includes the minimum, the ﬁrst
quartile, the median, the third quartile, and the maximum. The minimum of a
data set is the least value. The maximum is the greatest value.
2
EXAMPLE
Finding the Five Number Summary
Find the ﬁve number summary for the data below.
125 80 140 135 126 140 350
Step 1 Arrange the data in order from least to greatest.
80 125 126 135 140 140 350
Step 2 Find the minimum, maximum, and median.
80 125 126 135 140 140 350
The minimum is 80. The maximum is 350. The median is 135.
Step 3 Find the ﬁrst quartile and third quartile.
ﬁrst quartile (Q1) = 125 1 126 = 125.5
2
third quartile (Q3) = 140 1 140 = 140
2
The ﬁve number summary is minimum = 80, Q1 = 125.5, median = 135, Q3 = 140,
and maximum = 350.
Quick Check
2 Find the ﬁve number summary for each set of data.
a. 95 85 75 85 65 60 100 105 75 85 75
b. 11 19 7 5 21 53
Data sets can be separated into smaller equal parts. Percentiles separate data
sets into 100 equal parts. The percentile rank of a score is the percentage of
scores that are less than or equal to that score.
3
EXAMPLE
Finding a Percentile Rank
Of 25 test scores, eight are less than or equal to 75. What is the percentile rank of a
test score of 75?
Write a ratio of the number of scores less than or equal to 75 compared to the total
number of test scores.
8
25
8 = 0.32
25
= 32%
Number of test scores less than or equal to 75
Total number of test scores
8
Rewrite 25
as a percent.
The percentile rank of 75 is 32.
Quick Check
3 a. Of the 25 test scores, 15 scores are less than or equal to 85. What is the
percentile rank of 85?
b. Critical Thinking Is it possible to have a percentile rank of 0? Explain.
Lesson NY-2 Quartiles and Box-and-Whisker Plots
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Box-and-Whisker Plots
A box-and-whisker plot is a graph that summarizes a data set along a number
line. The box represents the ﬁrst quartile, the median, and the third quartile of the
data. The whiskers on the left and right sides represent the minimum and maximum
values. Use the ﬁve number summary to construct a box-and-whisker plot.
Box-and-Whisker Plot
whisker
minimum
4
EXAMPLE
whisker
box
first
quartile
third maximum
quartile
median
Real-World
Problem Solving
Agriculture The table below shows United States crops harvested from 1992 to
2004. Find the ﬁve number summary and construct a box-and-whisker plot of
the data.
Step 1 Arrange the data in order from least
to greatest. Find the minimum,
maximum, and the median.
308 314 316 317 321 321 321
324 325 326 326 327 332
Step 2 Find the ﬁrst quartile and third quartile.
first quartile = 316 1 317 = 633 = 316.5
2
2
third quartile = 326 1 326 = 652 = 326
2
2
Crops Harvested
Acres
Acres
Year (millions) Year (millions)
1992
1993
1994
1995
1996
1997
1998
317
308
321
314
326
332
326
1999
2000
2001
2002
2003
2004
327
325
321
316
324
321
SOURCE: Statistical Abstract of the United States.
Step 3 Draw a number line. Construct the box-and-whisker plot. Mark the
values of the minimum, ﬁrst quartile, median, third quartile,
and maximum.
Crops Harvested (millions of acres)
290 300 310 320 330 340 350
Quick Check
4 Construct a box-and-whisker plot for the following distances of bird migrations
in thousands of miles.
5 2.5 6 8 9 2 1 4 6.2 18
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EXAMPLE
Using a Graphing Calculator
Use a graphing calculator to make a box-and-whisker plot of the data below.
Find the ﬁve number summary.
10 26 18 35 14 11 17 29 31 25 27 20 19 12 13 26
L1
L2
L3
1
25
27
20
19
12
13
26
1:L1
Plot1 Plot2 Plot3
On Off
Type:
Xlist: L1
Freq: 1
L1(16) = 26
Med=19.5
Step 1
Press
1: Edit . . .
.
Enter data under
L1. (To clear L1,
move the cursor
over L1 and press
,
.)
Step 2
Y= for
Press
STAT PLOTS.
Enter 1.
Highlight ON and
also Type:
box-and-whisker
plot. Press
.
Step 3
Press
and
enter 9. Press
and move the
cursor across the
box-and-whisker
plot to ﬁnd the ﬁve
number summary.
The ﬁve number summary is minimum = 10, Q1 = 13.5, median = 19.5, Q3 = 26.5,
maximum = 35.
Quick Check
5 Use a graphing calculator to ﬁnd the ﬁve number summary for the following set of
student test grades shown below.
98 92 76 84 93 82 74 68 85 91 77 83 94 97 72 88 70 84 87 82
The interquartile range is the difference between the ﬁrst and third quartiles.
In a box-and-whisker plot, this is the width of the box. You can use a parallel
box-and-whisker plot to compare two sets of data.
6
EXAMPLE
Comparing Data Sets
The box-and-whisker plot below shows average monthly rainfall for Miami and
New Orleans. Which city shows the greater range in average monthly rainfall?
Average Monthly Rainfall (millimeters)
20 40 60 80 100 120 140 160 180 200 220
Miami, Fla.
New Orleans, La.
Miami has the longer box-and-whisker plot, so it has the greater range.
Quick Check
6 What do the interquartile ranges tell you about the average monthly rainfall for
Miami and New Orleans?
Lesson NY-2 Quartiles and Box-and-Whisker Plots
NY 735
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EXERCISES
For more exercises, see Extra Skill and Word Problem Practice.
Practice and Problem Solving
A
Practice by Example
Example 1
GO for
Help
(page NY 732)
Example 2
(page NY 733)
Example 3
(page NY 733)
Example 4
(page NY 734)
Example 5
(page NY 735)
Find the median, the ﬁrst quartile, and the third quartile.
1. 12 10 11 7 9 10 5
2. 4.5 3.2 6.3 5.2 5 4.8 6 3.9 12
3. 55 53 67 52 50 49 51 52 52
4. 101 100 100 105 101 102 104
5. Find the ﬁve number summary for the set of data.
1 1 3 3 4 5 7 7 8 9 9 32 31 29 35 30 32 11
6. Of 20 scores, 19 are less than or equal to 10. Find the percentile rank of 10.
7. Gerry ranks 48th in a class of 120 students. What is his percentile rank?
8 Construct a box-and-whisker plot for the ages of the members of the art club.
13 18 16 14 15 17 15 16 16 15 16 14 14 17 19
9. Use a graphing calculator to draw a box-and-whisker plot for the data below.
Then ﬁnd the interquartile range.
16 18 59 75 30 34 25 49 27 16 21 58 71 19 50
Example 6
10. Compare the box-and-whisker plots below. What can you conclude?
(page NY 735)
Ages of U.S. Olympic Soccer Team Players
15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34
Men
Women
B
Apply Your Skills
For Exercises 11–16, use the data below. Find each measure.
1.8 2.5 3.9 4.6 4.7 4.8 4.8 4.9
11. median
12. ﬁrst quartile
13. minimum
14. maximum
15. third quartile
16. percentile rank of 4.6
17. Use the box-and-whisker plot below. What can you conclude about the
areas of state parks?
Areas of State Parks (acres)
100
200
300
400
500
600
For Exercises 18–21, use the graph below showing box-and-whisker plots for two
sets of data, A and B.
66
68
70
72
74
76
78
Data Set A
Data Set B
18. Which set of data has the greater range?
19. Which set of data has the lesser median?
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80
82
84
86
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20. Which set of data has the greater interquartile range?
21. Which set of data has the lesser minimum?
For Exercises 22–24, use the ﬁve number summary given below.
Minimum = 10
Q1 = 20
Median = 30
22. Find the interquartile range.
Q3 = 42
Maximum = 75
23. Find the range of the data.
24. Use the data to construct a box-and-whisker plot.
25. Social Studies The plots below compare the percents of the voting-age
population who said they registered to vote in U.S. elections to the percents
who said they voted. Which conclusion best reﬂects the data collected?
Percents of Population Who Registered and Voted, 1990–2000
40
50
60
70
Registered
Voted
A. The percent who voted was about 15% less than the percent who
registered.
B. The percent who voted was about half the percent who registered.
C. The percent who voted was equal to the percent who registered.
D. The percent who voted was about 15% more than the percent who
registered.
26. Error Analysis In a class of 250 students, Emily had the tenth highest grade
average. She computed her percentile rank as 4. What was her error?
C
Challenge
27. Reasoning Can you ﬁnd the mean, median, and mode of a set of data by
looking at a box-and-whisker plot? Explain.
REGENTS
Test Prep
Multiple Choice
28. Find the ﬁrst quartile and third quartile in the following data set.
17 20 30 19 20 18 25 28 31 23 17 29 31 33 28
A. 19 and 30
B. 17 and 25
C. 19.5 and 30.5
D. 17 and 33
29. Of 20 test scores, sixteen are less than or equal to 80. What is the percentile
rank of a test score of 80?
F. 16th
Short Response
G. 85th
H. 25th
J. 80th
30. Describe how you could ﬁnd the 75th percentile score in a set of 20 scores.
Mixed Review
Lesson 1-1
Deﬁne variables and write an equation to model each situation.
31. The total cost of the number of CDs times $5.00.
32. The perimeter of a square equals 4 times the length of the side.
Lesson 1-3
Decide whether each statement is true or false. If the statement is false, give the
counterexample.
33. All whole numbers are rational numbers.
34. The square root of a number is always smaller than the number.
Lesson NY-2 Quartiles and Box-and-Whisker Plots
NY 737