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6-5. Data Distributions . Objective Describe the central tendency of a data set. Measures of central tendency: mean , median , mode, and range. Mean: average (sum of the values divided by the number of values) .

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6-5

Data Distributions

  • Objective
  • Describe the central tendency of a data set.
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Measures of central tendency:

mean, median, mode, and range

Mean: average (sum of the values divided by the number of values)

Median: middle value when the values are in numerical order, or the mean of the two middle numbers if there are an even number of values

Mode: value or values that occur most often

Range:difference between the least and greatest values

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mean:

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

The weights in pounds of six members of a football team are 161, 156, 150, 156, 150, and 163. Find the mean, median, mode, and range of the data set.

Write the data in numerical order.

150, 150, 156, 156, 161, 163

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

156

median: 150, 150, 156, 156, 161, 163

Find the mean of the two middle values.

Median: 156.

Mode(s) occur most often

Modes: 150 and 156

Range is greatest – least

Range: 163 – 150 = 13

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You Try!Example 2

The weights in pounds of five cats are 12, 14, 12, 16, and 16. Find the mean, median, mode, and range of the data set.

Write the data in numerical order.

12, 12, 14, 16, 16

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

14

Find the middle value.

median: 12, 12, 14, 16, 16

Median: 14

Mode(s) occur most often

Mode: 12 and 16

Range is greatest – least

Range: 12 – 16 = 4

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A value that is very different from the other values is called an outlier.

Much different value

Outlier

Most of data

Mean

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Example 3: Determining the Effect of Outliers

Identify the outlier in the data set {16, 23, 21, 18, 75, 21}, and determine how the outlier affects the mean, median, mode, and range of the data.

Write the data in numerical order.

16, 18, 21, 21, 23, 75

The outlier is the value much greater or less than the rest.

The outlier is 75

With the outlier:

median:

The median is 21

16, 18, 21, 21, 23, 75

mode: 21 occurs twice

range: 75 – 16 = 59

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median:

The median is 21.

16, 18, 21, 21, 23

Example 3 Continued

Without the outlier:

mode: 21 occurs twice. It is the mode.

range: 23 – 16 = 7

Without outlier:

With outlier

Effect:

mean:

+9.2

29

19.8

median:

21

21

None

mode:

21

21

None

range:

59

7

+52

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3+21+24+24+27+30

6

You Try!Example 4

Identify the outlier in the data set {21, 24, 3, 27, 30, 24} and determine how the outlier affects the mean, median, mode and the range of the data.

3, 21, 24, 24, 27, 30

Write the data in numerical order.

The outlier is the value much greater or less than the rest.

The outlier is 3

With the outlier:

= 21.5

mean:

median:

The median is 24.

3, 21, 24, 24, 27, 30

mode:

24 occurs twice

range:

30 – 3 = 27

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21+24+24+27+30

= 25.2

mean:

5

median:

The median is 24.

21, 24, 24, 27, 30

You Try!Example 4 Continued

Without the outlier:

mode: 24 occurs twice. It is the mode.

range: 30 – 21 = 9

Without outlier:

With outlier

Effect:

mean:

–3.7

21.5

25.2

median:

24

24

None

mode:

24

24

None

range:

27

9

+18

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As you can see, an outlier can strongly affect the mean of a data set, having little or no impact on the median and mode.

Therefore, the mean may not be the best measure to describe a data set that contains an outlier.

In such cases, the median or mode may better describe the center of the data set.

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Example 5: Choosing a Measure of Central Tendency

Rico scored 74, 73, 80, 75, 67, and 54 on six history tests. Use the mean, median, and mode of his scores to answer each question.

mean ≈ 70.7 median = 73.5 mode = none

A. Which measure best describes Rico’s scores?

Median: 73.5

Because the outlier of 54 lowers the mean

B. Which measure should Rico use to describe his test scores to his parents? Explain.

Median: 73.5

Median is greater than the mean

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You Try!Example 6

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 measure describes the score Josh received most often?

Mode: 75

B. Which measure best describes Josh’s scores? Explain.

Median: 81

median is greater than either the mean or the mode.

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Lesson Quiz: 6-5

1. The data set gives the times of Tara’s one-way ride to school (in minutes) for one week. Find the mean, median, mode, and range of the data set.

{8, 3, 5, 4, 5}

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

3. Which value best describes Tara’s ride time? Explain.

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Classwork/Homework

6-5Worksheet

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