Sampling Distribution of the Mean and Standard Deviation in Various Populations

The sampling distributions for the sample mean and the sample standard deviation are explored for a finite population, as well as several infinite populations (normal, uniform, Laplace, gamma). All distributions are chosen to have the same mean, , and standard deviation, , as the finite population. The finite population consists of the marks of 256 students on a midterm test. The population distribution is shown in the top plot. The bottom plot shows normal approximation to the sampling distribution (exact for the sample mean if the population is also normal) and its empirical estimate using the histogram. These histograms are available sample sizes, and for 1000, 5000 and 10000 samples at a time.
From the central limit theorem, the sampling distribution of the sample mean is approximated by for infinite populations. In the finite population case, the standard deviation is reduced slightly due to the finite-sample correction factor (see the Details section).
The sampling distribution for the sample standard deviation is more complicated, but as increases it is also approximated by a normal distribution. However, in this case, we need to use simulation to estimate the parameters in the approximating normal sampling distribution.
In all cases, as increases, the sampling distribution becomes more concentrated about the true parameter and the normal distribution approximation improves.
It is interesting that in the case of the mean, all sampling distributions converge to the same distribution as expected by the central limit theorem. But the sampling distribution for the standard deviation is very complex analytically for non-normal populations. Our computer experiments illustrate that it is approximately normal. Notice these distributions, although all approximately normal, depend on the population distribution.
The sensitivity of the results to random variation may be explored by varying the random seed.

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DETAILS

In the infinite population case, the sample mean has a normal distribution with mean 76 and standard deviation , where is the sample size.
Sampling in the finite population is without replacement. In this case, the variance of the sampling distribution for the mean is adjusted by multiplying by the finite population correction factor , where is the population size.
[1] I. McLeod, "Simple Random Sampling," Encyclopedia of Statistical Sciences, Vol. 12, 2nd ed. (S. Kotz, C. B. Read, N. Balakrishnan, and B. Vidakovic, eds.), New York: Wiley, 2005 p. 7740.
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