Sampling Distributions

Sampling distributions, unlike population and sample distributions, are theoretical; that is, they do not exist as empirical realities. We have already worked with theoretical distributions in the form of the binomial and standard normal distributions. Sampling distributions are theoretical because they are based on the notion of multiple (even infinite) samples being drawn from a single population. What sets sampling distributions apart from empirical distributions is that sampling distributions are created not from raw scores but, rather, from sample statistics. These descriptors can be means, proportions, or any other statistic. Imagine plotting the means in Table 7.2 to form a histogram.

Not terribly impressive, is it? Definitely leaves something to be desired. That is because there are only five samples. Sampling distributions start to take shape only when many samples have been drawn. If we continue the iterative process of drawing a sample, computing and plotting mean, throwing the sample back, and pulling a new sample, the distribution gradually starts looking something like the curve.

Now the distribution has some shape! It looks much better. It is, moreover, not just any old shape—it is a normal curve. What you have just seen is the central limit theorem (CLT) in action. The CLT states that any time descriptive statistics are computed from an infinite number of large samples, the resulting sampling distribution will be normally distributed. The sampling distribution clusters around the true population mean, and if you were to compute the mean of the sampling distribution (i.e., the mean of means), the answer you obtained would match the true population mean. Its standard deviation (called the standard error) is smaller than the population standard deviation because there is less dispersion, or variability, in means than in raw scores.