7.3 Comments on the Sampling Distribution of the Mean
We next turn to a few observations pertaining to the sampling distribution of the mean. The basis for the following set of comments is Table 7.2:
1. Each of the sample means is approximately equal to the population mean (μ = 3).
2. The sample means cluster much more closely about the population mean than do the original values of X, that is, V(
) = 0.33 < V(X) = 2.
) = 0.33 < V(X) = 2.
3. If larger samples were taken, their means would cluster even more closely about the population mean than before. (To see this, let n = 4. Then
Clearly, V(
) has decreased in value from 0.33.) In this regard, we may conclude that the larger the sample, the closer the sample mean is likely to be to the population mean. This property of the sample mean is referred to as consistency, that is, 
is a consistent estimator of μ if for large samples the probability is close to 1 that will be near μ or
) has decreased in value from 0.33.) In this regard, we may conclude that the larger the sample, the closer the sample mean is likely to be to the population mean. This property of the sample mean is referred to as consistency, that is, is a consistent estimator of μ if for large samples the probability is close to 1 that will be near μ or
So as n increases without bound, the sampling distribution of the mean ...
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