6.7 Use Measures of Central Tendency and Dispersion

Vocabulary Mean: average, “x-bar” Median: middle # when values written in numerical order; avg. of 2 numbers if there is an even set of data Mode: value that occurs most often Measure of dispersion: describes the spread of data

Vocabulary Range: difference of greatest value and least value Deviation from the mean: difference of the data value and the mean Mean absolute deviation: avg. deviation of the data from the mean

Compare measures of central tendency The heights (in feet) of 8 waterfalls in the state of Washington are listed below. Find mean, median, & mode. Which measure of central tendency best represents the data? 1000, 1000, 1181, 1191, 1200, 1268, 1328, 2584 SOLUTION x = = 10,752 8 = 1344

Compare measures of central tendency ANSWER The median best represents the data. The mode is significantly less than most of the data, and the mean is significantly greater than most of the data. The median is the mean of the two middle values, 1191 and 1200, or The mode is , 1000, 1181, 1191, 1200, 1268, 1328, 2584

Compare measures of dispersion The top 10 finishing times (in seconds) for runners in two men’s races are given. The times in a 100 meter dash are in set A, and the times in a 200 meter dash are in set B. Compare the spread of the data for the two sets using (a) the range and (b) the mean absolute deviation. A: 10.62, 10.94, 10.94, 10.98, 11.05, 11.13, 11.15, 11.28, 11.29, RUNNING B: 21.37, 21.40, 22.23, 22.23, 22.34, 22.34, 22.36, 22.60, 22.66, 22.73

SOLUTION a. A: – B: – = 0.7 = 1.36 The range of set B is greater than the range of set A. So, the data in B cover a wider interval than the data in A. ANSWER A: 10.62, 10.94, 10.94, 10.98, 11.05, 11.13, 11.15, 11.28, 11.29, 11.32

b. The mean of set A is 11.07, so the mean absolute deviation is: | – | + | – | + … + | – | 10 = The mean of set B is , so the mean absolute deviation is: | – | + | – | + … + | – | 10 =

The mean absolute deviation of set B is greater, so the average variation from the mean is greater for the data in B than for the data in A. ANSWER

RUNNING 2. The top 10 finishing times (in seconds) for runners in a men’s 400 meter dash are 46.89, 47.65, 48.15, 49.05, 49.19, 49.50, 49.68, 51.09, 53.31, and Compare the spread of the data with that of set A in Example 2 using (a) the range and (b) the mean absolute deviation. a. The range of finishing times for the men’s 400m dash (6.79) is greater than the range for set A (0.7). ANSWER b. The mean absolute deviation is greater for the men’s 400m dash (1.7246) than it is for set A (0.164).