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Sampling distributions are at the cornerstone of statistical inference. The sampling distribution of a statistic is a theoretical probability distribution of that statistic. As defined by Degroot and Schervish (2002), “the sampling distribution of a statistic tells us what values a statistic is likely to assume and how likely it is to assume those values prior to observing our data” (p. 391).

As an example, we will generate a theoretical sampling distribution of the mean for a given population with mean μ and variance, σ². The distribution we will create is entirely idealized in that it does not exist in nature anywhere. It is simply a statistical theory of how the distribution of means might look if we were able to take an infinite number of samples of a given size from a given population, and on each of these samples, calculate the sample mean statistic.

When we derive sampling distributions for a statistic, we are asking the following question:

If we were to draw an infinite number of samples of size nfrom this population and calculate the sample mean on each sample, what would the distribution of sample means look like?

If we can specify this distribution, then we can evaluate obtained sample means relative to it. That is, we will be able to compare our obtained means (i.e., the ones we obtain in real empirical research) to the theoretical sampling distribution of means, and answer the question:

If my obtained sample mean really did come from this population, what is the probability of obtaining a mean such as this?

If the probability is low, you might then decide to reject the assumption that the sample mean you obtained arose from the population in question. It could have, to be sure, but it probably did not. For continuous measures, our interpretation above is slightly informal, since the probability of any particular value of the sample mean in a continuous distribution is essentially equal to 0 (i.e., in the limit, the probability equals 0). Hence, the question is usually posed such that we seek to know the probability of obtaining a mean such as the one we obtained or more extreme.