110 Chapter 3 Numerical Summaries of Data Check Your Understanding 2. Find the sample variance and standard deviation for the following samples. a.2 2 1 5 4 5 0 1 b. 22 27 13 53 47 50 3. Find the population variance and standard deviation for the population values 7, 3, 2, 5, 4, 9, 6, 3. Answers are on page 123. Recall: A statistic is resistant if its value is not affected much by extreme data values. The standard deviation is not resistant Example 3.15 shows that the standard deviation is not resistant. EXAMPLE 3.15 Show that the standard deviation is not resistant In Example 3.14, we found that the sample standard deviation for six battery lifetimes was s = 1.414. The six lifetimes were 3, 4, 6, 5, 4, and 2. Assume that the battery with a lifetime of 6 actually had a lifetime of 20, so that the lifetimes are 3, 4, 20, 5, 4, and 2. Compute the standard deviation. Solution Following is the display from the TI-84 Plus. Including the extreme value of 20 in the data set has increased the sample standard deviation from 1.414 to 6.772. Clearly, the standard deviation is not resistant. Objective 4 Approximate the standard deviation with grouped data Approximating the Standard Deviation with Grouped Data Sometimes we don’t have access to the raw data in a data set, but we are given a frequency distribution. In Section 3.1, we learned how to approximate the sample mean from a frequency distribution. We now show how to approximate the standard deviation. We useTable 3.6 to illustrate the method.Table 3.6 presents the number of text messages sent via cell phone by a sample of 50 high school students. In Section 3.1, we computed the approximate sample mean to be ¯x = 137. We will now approximate the standard deviation. Table 3.6 Number of Text Messages Sent by High School Students Number of Text Messages Sent Frequency 0–49 10 50–99 5 100–149 13 150–199 11 200–249 7 250–299 4
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