278 CHAPTER 6 Normal Probability Distributions Population standard deviation: s = Ea1x - m2 2 N Population variance: s 2 = a1x - m2 2 N Caution: Because the calculations are typically performed with software or calculators, be careful to correctly distinguish between the variance of a sample and the variance of a population. Behavior of Sample Variances 1. The distribution of sample variances tends to be a distribution skewed to the right. 2. The sample variances target the value of the population variance. (That is, the mean of the sample variances is the population variance. The expected value of the sample variance is equal to the population variance.) Sampling Distribution of the Sample Variance EXAMPLE 4 Consider repeating this process: Roll a die 5 times and find the variance s 2 of the results. What do we know about the behavior of all sample variances that are generated as this process continues indefinitely? YOUR TURN. Do Exercise 14 “Sampling Distribution of the Variance.” SOLUTION Figure 6-20 illustrates a process of rolling a die 5 times and finding the variance of the results. Figure 6-20 shows results from repeating this process 10,000 times, but the true sampling distribution of the sample variance involves repeating the process indefinitely. Because the values of 1, 2, 3, 4, 5, 6 are all equally likely, the population has a variance of s 2 = 2.9, and the 10,000 sample variances included in Figure 6-20 have a mean of 2.9. If the process is continued indefinitely, the mean of the sample variances will be 2.9. Also, Figure 6-20 shows that the distribution of the sample variances is a skewed distribution, not a normal distribution with its characteristic bell shape. Variances Sample 1 Sample 2 Sample 3 (Population Variance is s2 5 2.9) Distribution of Sample Variances Sample Variances Sample s2 5 2.9 Sample variances tend to have a skewed distribution • • • 1.8 2.3 2.2 Sampling Procedure: Roll a die 5 times and find the variance s2 for each sample. FIGURE 6-20 Sample Variances from 10,000 Trials CAUTION When working with population standard deviations or variances, be sure to evaluate them correctly. In Section 3-2 we saw that the computations for population standard deviations or variances involve division by the population size N instead of n - 1, as shown here.
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