798 CHAPTER 12 Statistics Measures of dispersion are used to measure the variability of the data, including the spread of the data, and how the data vary about the mean. Consider two sets of data: 6, 6, 6, 6, 6, 6, 6, 6 and 2, 4, 6, 6, 6, 6, 8, 10. The mean, median, mode, and midrange for both sets of data are all equal to 6. However, these are very different sets of data. Measures of dispersion help us to have a better understanding of a set of data. The range and standard deviation are the measures of dispersion that will be discussed in this section. Range The range is the difference between the highest and lowest values; it indicates the total spread of the data. Why This Is Important Knowing the measures of central tendency of a set of data is important, but knowing the measures of dispersion is just as important. As we will see in this section, measures of dispersion help us better understand the data and help us draw accurate conclusions about the data. Range The range of a set of data can be calculated using the following formula. = − Range highest value lowest value Example 1 Determine the Range The number of milligrams of caffeine in seven selected energy drinks are given below. Determine the range of these data. 160, 200, 300, 200, 140, 80, 150 Solution = − = − = Range highest value lowest value 300 80 220. The range of the number of milligrams of caffeine is 220. 7 Now try Exercise 7 Standard Deviation The second measure of dispersion we discuss in this section, the standard deviation, measures how much the data differ from the mean. It is symbolized either by the letter s or by the Greek lowercase letter sigma, σ. The s is used when the standard deviation of a sample is calculated. The σ is used when the standard deviation of the entire population is calculated. Since we are assuming that all data presented in this section are for samples, we use s to represent the standard deviation (note, however, that on the height and weight charts on page 804, σ is used). The larger the variability of the data about the mean, the larger the standard deviation is. Consider the following two sets of data. 5, 8, 9, 10, 12, 13 8, 9, 9, 10, 10, 11 Both have a mean of 9.5. Which set of values on the whole do you believe differs less from the mean of 9.5? Fig 12.16, which shows a visual picture of the two sets of data, may make the answer more apparent. The scores in the second set of data are closer 5 6 7 8 9 10 11 12 13 9.5 x x x x 5 6 7 8 9 10 11 12 13 9.5 x x x x x x x x Figure 12.16 UfaBizPhoto/Shutterstock
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