3-2 Measures of Variation 107 EXAMPLE 2 Calculating Standard Deviation with Formula 3-4 Use Formula 3-4 to find the standard deviation of these “Space Mountain” wait times (minutes) from Example 1: 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20. SOLUTION The left column of Table 3-4 on the next page summarizes the general procedure for finding the standard deviation using Formula 3-4, and the right column illustrates that procedure for the specific sample values 50, 25, 75, 35, 50, 25, 30, 50, 45, 25, 20. The result shown in Table 3-4 is 16.6 minutes, which is rounded to one more decimal place than is present in the original list of sample values. Also, the units for the standard deviation are the same as the units of the original data. Because the original data all have units of minutes, the standard deviation is 16.6 minutes. YOUR TURN. Find the standard deviation in Exercise 7 “Celebrity Net Worth.” CP FORMULA 3-4 s = CΣ1x - x22 n - 1 sample standard deviation FORMULA 3-5 s = Cn1Σx22 - 1Σx22 n1n - 12 shortcut formula for sample standard deviation (used by calculators and software) Later we give the reasoning behind these formulas, but for now we recommend that you use Formula 3-4 for an example or two, and then learn how to find standard deviation values using a calculator or software. Important Properties of Standard Deviation ■ The standard deviation is a measure of how much data values deviate away from the mean. ■ The value of the standard deviation s is never negative. It is zero only when all of the data values are exactly the same. ■ Larger values of s indicate greater amounts of variation. ■ The standard deviation s can increase dramatically with one or more outliers. ■ The units of the standard deviation s (such as minutes, feet, pounds) are the same as the units of the original data values. ■ The sample standard deviation s is a biased estimator of the population standard deviation s, which means that values of the sample standard deviation s do not center around the value of s. (This is explained in Part 2.) Example 2 illustrates a calculation using Formula 3-4 because that formula better illustrates that the standard deviation is based on deviations of sample values away from the mean. More Stocks, Less Risk In their book Investments, authors Zvi Bodie, Alex Kane, and Alan Marcus state that “the average standard deviation for returns of portfolios composed of only one stock was 0.554. The average portfolio risk fell rapidly as the number of stocks included in the portfolio increased.” They note that with 32 stocks, the standard deviation is 0.325, indicating much less variation and risk. They make the point that with only a few stocks, a portfolio has a high degree of “firm-specific” risk, meaning that the risk is attributable to the few stocks involved. With more than 30 stocks, there is very little firm-specific risk; instead, almost all of the risk is “market risk,” attributable to the stock market as a whole. They note that these principles are “just an application of the well-known law of averages.”
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