106 Chapter 3 Numerical Summaries of Data deviations, while those with less spread will have smaller squared deviations. The average of the squared deviations is the population variance. Explain It Again Another formula for the population variance: An alternate formula for the population variance is σ2 = x2 − Nμ2 N This formula always gives the same result as the one in the definition. DEFINITION Let x1, ..., xN denote the values in a population of size N. Let μ denote the population mean. The population variance, denoted by σ2, is σ2 = (x − μ)2 N We present the procedure for computing the population variance. Procedure for Computing the Population Variance Step 1: Compute the population mean μ. Step 2: For each population value x, compute the deviation x − μ. Step 3: Square the deviations, to obtain quantities (x − μ)2. Step 4: Sum the squared deviations, obtaining(x − μ)2. Step 5: Divide the sum obtained in Step 4 by the population size N to obtain the population variance σ2. CAUTION The population variance will never be negative. It will be equal to zero if all the values in a population are the same. Otherwise, the population variance will be positive. In practice, variances are usually calculated with technology. It is a good idea to compute a few by hand, however, to get a feel for the procedure. EXAMPLE 3.11 Computing the population variance Compute the population variance for the San Francisco temperatures. Solution The calculations are shown in Table 3.4. Step 1: Compute the population mean: μ = 51 + 54 + 55 + 56 + 58 + 60 + 60 + 61 + 63 + 62 + 58 + 52 12 = 57.5 Step 2: Subtract μ from each value to obtain the deviations x − μ. These calculations are shown in the second column of Table 3.4. Table 3.4 Calculations for the Population Variance in Example 3.11 x x−μ (x−μ)2 51 −6.5 (−6.5)2 = 42.25 54 −3.5 (−3.5)2 = 12.25 55 −2.5 (−2.5)2 = 6.25 56 −1.5 (−1.5)2 = 2.25 58 0.5 0.52 = 0.25 60 2.5 2.52 = 6.25 60 2.5 2.52 = 6.25 61 3.5 3.52 = 12.25 63 5.5 5.52 = 30.25 62 4.5 4.52 = 20.25 58 0.5 0.52 = 0.25 52 −5.5 (−5.5)2 = 30.25 μ = 57.5 (x − μ)2 = 169 σ2 = 169 12 = 14.083
navidi_monk_essential_statistics_1e_ch1_3
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