The original central limit theorem (CLT) states that under certain conditions, the mean of a random sample of

numbers has an approximately normal (Gaussian) distribution as the sample size

approaches infinity. According to theorems proven by Robbins [1] and Billingsley [2] (see [3]), normality will be approached even if the sample size

is random, provided that it is sufficiently large and its distribution sufficiently narrow. This Demonstration is intended to show that the means of samples of uniformly distributed random numbers, of random sample sizes, indeed have an approximately normal distribution if the sample sizes are sufficiently large and picked from a uniform distribution, and that they become closer to being normally distributed if the sample size range narrows.

You can select with sliders both the size limits,

and

, of the random samples, and the range,

and

, from which the random numbers are to be chosen. The program then generates samples of random numbers within the specified limits and plots the histogram of the mean values of these samples. It calculates a corresponding normal distribution having the same mean,

, and standard deviation,

, as those of the generated means, and plots its probability density function superimposed on the histogram. For comparison, the program also calculates the analytical mean,

, and standard deviation,

, of the normal distribution had the sets been of uniform of size

. It plots the corresponding Q-Q plot for visual inspection of the normality of the generated data's distribution.

[1] H. E. Robbins, "The Asymptotic Distribution of the Sum of a Random Number of Random Variables,"

*Bulletin of the American Mathematical Society*,

**54**(12), 1948 pp. 1151–1161.

projecteuclid.org/euclid.bams/1183513324.

[3] W. Feller,

*An introduction to Probability Theory and Its Application,* 2nd ed., Vol. II, New York: John Wiley & Sons, 1971.