The Central Limit Theorem: Why the Sample Mean Becomes Normal and Narrows as the Sample Grows

Abstract

When a quantity is estimated by averaging a sample, the average itself isuncertain, varying from one sample to another, so it has a distribution of itsown; the central limit theorem says that this distribution of the sample mean becomes a normal, bell-shaped one as the sample grows, almost whatever the distribution of the individual values, and narrows as the sample grows, which is why the average of a large sample is both predictable in shape and close to the true value. This study examines the central limit theorem and why the sample mean becomes normal and narrows as the sample grows. The distribution of the sample mean was examined as the size of the sample was increased, for samples drawn from a markedly non-normal population, so that the change in the shape and the spread of that distribution with sample size could be seen. As the sample grew, the distribution of the sample mean changed in two ways: it became more nearly normal in shape, losing the skewness of the population from which the samples were drawn, so that even averages of a strongly non-normal quantity were bell-shaped once the sample was moderately large; and it became narrower, its spread shrinking as the sample grew, in proportion to one over the square root of the sample size, so that the sample mean clustered ever more tightly about the true population mean. The distribution of the sample mean for a large sample was therefore a narrow normal curve centred on the true mean, regardless of the shape of the population. This is why so much of statistics rests on the normal distribution and why larger samples give more reliable estimates, and it underlies confidence intervals and tests built on the sample mean. The study shows that the sample mean becomes normal and narrows as the square root of the sample size, and it discusses the implications for statistical estimation and inference.  KEYWORDS: Central limit theorem, Sample mean, Sampling distribution,Normal distribution, Standard error, Sample size, Statistical inference,Probability, Estimation