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) 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

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. mean: 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

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

A value that is very different from the other values is called an outlier. Much different value Outlier Most of data Mean

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: 16, 18, 21, 21, 23, 75 The median is 21 mode: 21 occurs twice range: 75 – 16 = 59

Example 3 Continued Without the outlier: 16, 18, 21, 21, 23 median: The median is 21. mode: 21 occurs twice. It is the mode. range: 23 – 16 = 7 With outlier Without outlier: Effect: 29 mean: 19.8 +9.2 21 median: 21 None 21 mode: 21 None 59 range: 7 +52

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. Write the data in numerical order. 3, 21, 24, 24, 27, 30 The outlier is the value much greater or less than the rest. The outlier is 3 With the outlier: 3+21+24+24+27+30 6 mean: = 21.5 median: 3, 21, 24, 24, 27, 30 The median is 24. mode: 24 occurs twice range: 30 – 3 = 27

You Try! Example 4 Continued Without the outlier: mean: 21+24+24+27+30 5 = 25.2 21, 24, 24, 27, 30 median: The median is 24. mode: 24 occurs twice. It is the mode. range: 30 – 21 = 9 With outlier Without outlier: Effect: 21.5 mean: 25.2 –3.7 24 median: 24 None 24 mode: 24 None 27 range: 9 +18

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.

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

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.

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.

Classwork/Homework 6-5 Worksheet