Sampling Distribution

Example 1: Finite Population

Example 1 (created by author for this article) contains N = 6 items (presented in Table 1), and suppose we are interested in a sample size of 3, n = 3, so there are

unique random samples from the data set. The population median is 6, the population mean is 5.8, and the population standard deviation is 2.4.

Table 2 contains the 20 unique samples as well as the values for the following statistics: sample mean, sample median, and sample standard deviation. Figure 2 shows [Page 933]the sampling distributions of the sample mean, sample median, and sample standard deviation. Notice that the sample distributions are centered on the associated parameter that is, the sampling distribution of the median is centered on 6, and the same is true for the mean (centered at 5.8) and standard deviation (centered at 2.4). Also, notice that there is variability in the sampling distributions. This variability is known as sampling variability—the variability that is induced by the act of taking a random sample. The value of a statistic does not equal the value of the parameter (typically) but varies around the parameter.

By looking at the sampling distribution, one can determine if an observed statistic is rare or common. For example, suppose one takes a random sample of three items and finds that the sample mean is 4 or below. If the random sample is from the original population, then the chance of observing an x equalto4or less is 1/20 = 0.05. So there is only a 5% chance that the random sample is from the original population.

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