A Large-Scale Sampling Study of the Central Limit Effect

Abstract

For each of the values N = 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, ten thousand samples of N observations were drawn from a long-tailed, roughly L-shaped population and their standardized means were calculated. Both frequency and cumulative frequency distributions were plotted for the entire sampling distribution of the 10,000 values of Z = (X̄ - μ)/(σ/√N) obtained at each value of N. For comparison the “approached” normal distribution was also shown in each figure. As a check on the randomness of sampling and the extent of sampling error, the entire procedure was simultaneously “replicated” with observations drawn from an essentially normal population, using the same pseudorandom numbers to identify observations to be drawn. The resulting “control” distributions of means behaved as they should if sampling were random and sampling error unexceptional, coinciding closely with the “approached” normal distributions. For samples from the L-shaped population it was not until N = 32 that the frequency distribution of X̄ ceased to be clearly multimodal, although it was still quite appreciably skewed with a conspicuous deficit from normality at the left tail and surplus at the right. The latter effects slowly diminished until at N = 256 they were no longer apparent from casual inspection of the histograms. However, they were obvious, at all N's, in graphs showing cumulative distributions for the testing tails: ordinatewise departures from normality were quite appreciable, relative to both sampling error and to the corresponding ordinate of the “approached” normal ogive.