Sampling distributions

One of the most important concepts in inferential statistics is that of the sampling distribution.

That's because the use of sampling distributions is what allows us to make "probability" statements in inferential statistics.

• A sampling distribution is defined as "The theoretical probability distribution of the values of a statistic that results when all possible random samples of a particular size are drawn from a population." (For simplicity you can view the idea of "all possible samples" as taking a million random samples. That is, just view it as taking a whole lot of samples!)

• A one specific type of sampling distribution is called the sampling distribution of the mean. If you wanted to generate this distribution through the laborious process of doing it by hand (which you would NOT need to do in practice), you would randomly select a sample, calculate the mean, randomly select another sample, calculate the mean, and continue this process until you have calculated the means for all possible samples. This process will give you a lot of means, and you can construct a line graph to depict your sampling distribution of the mean.

• The sampling distribution of the mean is normally distributed (as long as your sample size is about 30 or more for your sampling).

• Also, note that the mean of the sampling distribution of the mean is equal to the population mean! That tells you that repeated sampling will, over the long run, produce the correct mean. The spread or variance shows you that sample means will tend to be somewhat different from the true population mean in most particular samples.

contributed by Karen Burke, EdD

It is important to understand that researchers do not actually empirically construct sampling distributions!

When conducting research, researchers typically select only one sample from the population of interest; they do not collect all possible samples.

• The computer program that a researcher uses (e.g., SPSS and SAS) uses the appropriate sampling distribution for you.

• The computer program will look at the type of statistical analysis you select (and also consider certain additional information that you have provided, such as the sample size in your study), and then the statistical program selects the appropriate sampling distribution.

contributed by Karen Burke, EdD