Section 3.2 Measures of Spread 107 Step 3: Square the deviations. These calculations are shownin the third column ofTable 3.4. Step 4: Sum the squared deviations to obtain (x − μ)2 = 169 Step 5: The population size is N = 12. Divide the sum obtained in Step 4 by N to obtain the population variance σ2. σ2 = (x − μ)2 N = 169 12 = 14.083 In Example 3.11, note how important it is to make all the deviations positive, which we do by squaring them. If we simply add the deviations without squaring them, the positive and negative ones will cancel each other out, leaving 0. Check Your Understanding 1. Compute the population variance for the St. Louis temperatures. Compare the result with the variance for the San Francisco temperatures, and interpret the result. Answer is on page 123. The sample variance When the data values come from a sample rather than a population, the variance is called the sample variance. The procedure for computing the sample variance is a bit different from the one used to compute a population variance. CAUTION The sample variance will never be negative. It will be equal to zero if all the values in a sample are the same. Otherwise, the sample variance will be positive. DEFINITION Let x1, ..., xn denote the values in a sample of size n. The sample variance, denoted by s2, is s2 = (x − ¯x)2 n − 1 The formula is the same as for the population variance, except that we replace the population mean μ with the sample mean ¯x, and we divide by n − 1 rather than N. Explain It Again Another formula for the sample variance: An alternate formula for the sample variance is s2 = x2 − nx¯2 n − 1 This formula will always give the same result as the one in the definition. We present the procedure for computing the sample variance. Procedure for Computing the Sample Variance Step 1: Compute the sample mean ¯x. Step 2: For each sample value x, compute the difference x − ¯x. This quantity is called a deviation. Step 3: Square the deviations, to obtain quantities (x − ¯x)2. Step 4: Sum the squared deviations, obtaining(x − ¯x)2. Step 5: Divide the sum obtained in Step 4 by n − 1 to obtain the sample variance s2. EXAMPLE 3.12 Computing the sample variance A company that manufactures batteries is testing a new type of battery designed for laptop computers. They measure the lifetimes, in hours, of six batteries, and the results are 3, 4, 6, 5, 4, and 2. Find the sample variance of the lifetimes.
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