Sampling Distributions

Consider a random sample selected from a population that is used to make an inference about some population characteristic, such as the population mean, , using a sample statistic, such as the sample mean, . The inference is based on the realization that every random sample has a different number for , and, thus,  is a random variable. The sampling distribution of this statistic is the probability distribution of the sample means obtained from all possible samples of the same number of observations drawn from the population.

We illustrate the concept of a sampling distribution by considering the position of supervisor with six employees, whose years of experience are

2

4

6

6

7

8

Two of these employees are to be chosen randomly for a particular work group. The mean of the years of experience for this population of six employees is

Now, let us consider the mean number of years of experience of the two employees chosen randomly from the population of six. Fifteen () possible different random samples could be selected. Table 1 shows all of the possible samples and associated sample means.
Table1: Samples and sample means from the worker population sample size n = 2.
Sample    Sample mean    Sample    Sample mean
2,4                  3.0                 4, 8                6.0
2.6                 4.0                  6,6                 6.0
2,6                 4.0                  6,6                 6.5
2,7                 4.5                  6,8                 7.0
2,8                 5.0                  6,7                 6.5
4,6                5.0                   6,8                 7.0
4,6                5.0                   7,8                 7.5
4,7                5.5     
Each of the 15 samples in Table 1 has the same probability, 1/15, of being selected. Note that there are several occurrences of the same sample mean. For example, the sample mean 5.0 occurs three times, and, thus, the probability of obtaining a sample 5.0 is 3/15. Table 2 represents the sampling distribution for the various sample means from the population, and the probability function is graphed in Figure 1.
Table 2: Sampling distribution of the sample means from the worker population sample size n = 2.

Sample mean    Probability of

        3.0                 1/15
        4.0                 2/15
        4.5                 1/15
        5.0                 3/15
        5.5                 1/15
        6.0                 2/15
        6.5                 2/15
        7.0                 2/15
        7.5                 1/15

We see that, while the number of years of experience for the six workers ranges from 2 to 8, the possible values of the sample mean have a range from only 3.0 to 7.5. In addition, more of the values lie in the central portion of the range.

Table 3 presents similar result for a sample of size n = 5 for sampling distribution. Notice that the means are concentrated over a narrow range. These sample means are all close to the population mean =5.5.  We will always find this to be true—the sampling distribution becomes concentrated closer to the population mean as the sample size increases. This is the important result provides an important foundation for statistical inference.
Table3: Samples and sample means from the worker population sample size n = 5.
Sample           Means       Probability
2,4,6,6,7           5.0          1/6
2,4,6,6,8           5.2          1/6
2,4,6,7,8           5.4          1/6
2,6,6,7,8           5.8          1/6
4,6,6,7,8           6.2          1/6