algebra unit 9.3

UNIT 9.3 MEASURES OF CENTRALUNIT 9.3 MEASURES OF CENTRALTENDENCYTENDENCY

Warm UpSimplify each expression.

1. 2. 102 – 53

3. Use the data below to make a stem-and-leaf plot.

7, 8, 10, 18, 24, 15, 17, 9, 12, 20, 25, 18, 21, 12

60 49

Describe the central tendency of a data set.

Create box-and-whisker plots.

Objectives

mean quartilemedian interquartile range (IQR)mode box-and-whisker plotrangeoutlier

Vocabulary

A measure of central tendency describes how data clusters around a value.

• The mean is the sum of the values in the set divided by the number of values in the set.

• The median the middle value when the values are in numerical order, or the mean of the two middle values if there are an even number of values.

• The mode is the value or values that occur most often. There may be one mode or more than one mode. If no value occurs more often than another, we say the data set has no mode.

The range of a set of data is the difference between the least and greatest values in the set. The range describes the spread of the data.

Example 1A: Finding Mean, Median, Mode, and Range of a Data Set

Find the mean, median, mode, and range of the data set.

The number of hours students spent on a research project: 2, 4, 10, 7, 5

median: 2, 4, 5, 7, 10The median is 5.

range: 10 – 2 = 8

mode: none

Write the data in numerical order.

Add all the values and divide by the number of values.

There are an odd number of values. Find the middle value.

No value occurs more than once.

mean:

Example 1B: Finding Mean, Median, Mode, and Range of a Data Set

Find the mean, median, mode, and range of each data set.

The weight in pounds of six members of a basketball team: 161, 156, 150, 156, 150, 163

mean:



There are an even number of values. Find the mean of the two middle values.

median: 150, 150, 156, 156, 161, 163The median is 156.

Example 1B Continued

150, 150, 156, 156, 161, 163

modes: 150 and 156 150 and 156 both occur more often than any other value.range: 163 – 150 = 13

Check It Out! Example 1a Find the mean, median, mode, and range of the data set.

8, 8, 14, 6 Write the data in numerical order.


mean:

median: 6, 8, 8, 14The median is 8.

mode: 8

range: 14 – 6 = 8


8 occurs more than any other value.

Check It Out! Example 1b Find the mean, median, mode, and range of the data set.

1, 5, 7, 2, 3 Write the data in numerical

order.Add all the values and divide

by the number of values.

There are an odd number of values. Find the middle value.

No value occurs more than once.

median: 1, 2, 3, 5, 7The median is 3.

mode: none

range: 7 – 1 = 6

mean:

Check It Out! Example 1c Find the mean, median, mode, and range of the data set.

12, 18, 14, 17, 12, 18



median: 12, 12, 14, 17, 18, 18 The median is 15 .


mean:

Check It Out! Example 1c Continued

Find the mean, median, mode, and range of the data set.

mode: 12, 18 12 and 18 both occur more often than any other value.

range: 18 – 12 = 6

12, 12, 14, 17, 18, 18

A value that is very different from other values in the set is called an outlier. In the data below, one value is much greater than the other values. This causes the mean to be greater than all of the other data values. In this case, either the median or mode would better describe the data.

Example 2: Choosing a Measure of Central Tendency

Rico scored 74, 73, 80, 75, 67, and 55 on six history tests. Use the mean, median, and mode of his scores to answer each question.

mean ≈ 70.7 median = 73.5 mode = noneA. Which value gives Rico’s test average?

The average of Rico’s scores is the mean, 70.7.

B. Which values best describes Rico’s scores?Median; most of his scores are closer to 73.5 than to 70.6.

The mean is lower than most of Rico’s scores because he scored a 55 on one test. Since there is no mode, it is not a good description of the data.

Check It Out! Example 2

Josh scored 75, 75, 81, 84, and 85 on five tests. Use the mean, median, and mode of his scores to answer each question.

mean = 80 median = 81 mode = 75

a. Which value describes the score Josh received most often?Josh has two scores of 75 which is the mode.

b. Which value best describes Josh’s scores? Explain.The median best describes Josh’s scores. The mode is his lowest score, and the mean is lowered by the two scores of 75.

Measures of central tendency describe how data tends toward one value. You may also need to know how data is spread out across several values.

Quartiles divide a data set into four equal parts. Each quartile contains one-fourth of the values in the set. The interquartile range (IQR) is the difference between the upper and lower quartiles. The IQR represents the middle half of the data.

A box-and-whisker plot can be used to show how the values in a data set are distributed. The minimum is the least value that is not an outlier. The maximum is the greatest value that is not an outlier. You need five values to make a box-and-whisker plot: the minimum, first quartile, median, third quartile, and maximum.

Mathematically, any value that is 1.5(IQR) less than the first quartile or 1.5(IQR) greater than the third quartile is an outier.

Helpful Hint

Example 3: Sports Application

The number of runs scored by a softball team at 19 games is given. Use the data to make a box-and-whisker plot.

3, 8, 10, 12, 4, 9, 13, 20, 12, 15, 10, 5, 11, 5, 10, 6, 7, 6, 11

Step 1 Order the data from least to greatest.

3, 4, 5, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 13, 15, 20

Step 2 Identify the five needed values and determine whether there are any outliers.

Example 3 Continued

3, 4, 5, 5, 6, 6, 7, 8, 9, 10, 10, 10, 11, 11, 12, 12, 13, 15, 20

Q16

Q312

Q210

Minimum

3

Maximum

20

IQR: 12 – 6 = 6 1.5(6) = 9

6 – 9 = –3 12 + 9 = 21

No values are less than –3 or greater than 21, so there are no outliers.

Example 3 Continued

Half of the scores are between 6 and 12 runs per game. One-fourth of the scores are between 3 and 6. The greatest score earned by this team is 20.

0 8 16 24

Median

First quartile Third quartile

● ●●● ●

Minimum Maximum

An outlier is represented on a box-and-whisker plot by a point that is not connected to the box by whiskers.

Writing Math

Check It Out! Example 3

Use the data to make a box-and-whisker plot.

13, 14, 18, 13, 12, 17, 15, 12, 13, 19, 11, 14, 14, 18, 22, 23

Step 1 Order the data from least to greatest.

11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23

Step 2 Identify the five needed values and determine whether there are any outliers.

11, 12, 12, 13, 13, 13, 14, 14, 14, 15, 17, 18, 18, 19, 22, 23

Q113

Q318

Q214

Minimum

11

Maximum

23

Check It Out! Example 3 Continued

IQR: 18 – 13 = 5 1.5(5) = 7.5

13 – 7.5 = 5.5 18 + 7.5 = 25.5

No values are less than 5.5 or greater than 25.5, so there are no outliers.

Half of the data are between 13 and 18. One-fourth of the data are between 11 and 13. The greatest value is 23.

Check It Out! Example 3 Continued

8 16 24

Median

First quartile Third quartile

• ●•• •

Minimum Maximum

1. Find the mean, median, mode, and range of the data set.

Lesson Quiz: Part I

The number of hours Gerald mowed lawns in one week: 7, 3, 5, 4, 5

mean: 4.8; median: 5; mode: 5; range: 4

Lesson Quiz: Part II

The following list gives times of Tara’s one-way ride to school (in minutes) for one week: 12, 23, 13, 14, 13. Use the mean, median, and mode of her times to answer each question.

mean = 15 median = 13 mode = 13

2. Which value describes the time that occurred most often? mode, 13

3. Which value best describes Tara’s ride time? Explain.Median or mode: 13; 13 occurred twice, and most times are near this value.

Lesson Quiz: Part III

4. The number of inches of snow that fell during the last 8 winters in one city are given. Use the data to make a box-and-whisker plot.

25, 17, 14, 27, 20, 11, 29, 32

1115.5 22.5 28

32

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