112 CHAPTER 3 Describing, Exploring, and Comparing Data methods of statistics discussed in later chapters. The standard deviation has the advantage of using only algebraic operations. Because it is based on the square root of a sum of squares, the standard deviation closely parallels distance formulas found in algebra. There are many instances where a statistical procedure is based on a similar sum of squares. Consequently, instead of using absolute values, we square all deviations 1x - x 2 so that they are nonnegative, and those squares are used to calculate the standard deviation. Why Divide by n − 1? After finding all of the individual values of 1x - x 22 we combine them by finding their sum. We then divide by n - 1 because there are only n - 1 values that can assigned without constraint. With a given mean, we can use any numbers for the first n - 1 values, but the last value will then be automatically determined. With division by n - 1, sample variances s2 tend to center around the value of the population variance s 2; with division by n, sample variances s2 tend to underestimate the value of the population variance s 2. How Do We Make Sense of a Value of Standard Deviation? Part 1 of this section included the range rule of thumb for interpreting a known value of a standard deviation or estimating a value of a standard deviation. (See Examples 4 and 5.) Two other approaches for interpreting standard deviation are the empirical rule and Chebyshev’s theorem. Empirical (or 68-95-99.7) Rule for Data with a Bell-Shaped Distribution A concept helpful in interpreting the value of a standard deviation is the empirical rule. This rule states that for data sets having a distribution that is approximately bellshaped, the following properties apply. (See Figure 3-5.) ■ About 68% of all values fall within 1 standard deviation of the mean. ■ About 95% of all values fall within 2 standard deviations of the mean. ■ About 99.7% of all values fall within 3 standard deviations of the mean. 34.13% 34.13% 13.59% 13.59% 2.14% 2.14% 0.13% 0.13% 68% within 1 standard deviation 95% within 2 standard deviations 99.7% of data are within 3 standard deviations of the mean (x − 3s to x + 3s) x − s x + s x + 2s x + 3s x x − 2s x − 3s FIGURE 3-5 The Empirical Rule EXAMPLE 6 The Empirical Rule IQ scores have a bell-shaped distribution with a mean of 100 and a standard deviation of 15. What percentage of IQ scores are between 70 and 130?
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