Population, sampling, and sample distributions

Samples (bottom distributions) of any size, N, can be drawn from a theoretical population distribution (top). The statistical moments of each sample, such as its mean and standard deviation, can be calculated from the sample data. The moments of a sample distribution are referred to as statistics of the sample. The moments of a population distribution are referred to as parameters of the population.

If samples are drawn from the population with replacement, then any number of samples of a given size, N, can be drawn. The moments of these samples themselves form distributions. Consider, for example, the means of the samples drawn from the population. One could prepare a frequency distribution of these means. A distribution of a sample statistic is referred to as a sampling distribution of that statistic. The middle distribution in the figure above is the sampling distribution of the mean. It is possible to prepare a sampling distribution of any statistic that can be calculated from sample data. For example, one could prepare a sampling distribution of the sample standard deviations.

Sampling distributions are important because they can be used to calculate the probabilities of experimental outcomes under the null hypothesis.

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