Thayer Watkins
Silicon Valley
& Tornado Alley

The Distribution of Sample Means
as a Function of Sample Size

The purpose of this page is to illustrate that as the size of the samples increases the closer the distribution of sample means approaches that of a normal distribution. This purpose is accomplished by drawing 2000 samples, computing their means and constructing the histogram of those sample means. (Each time the screen is refreshed a new batch of 2000 samples is created.)

Let p(x) be the probability density function for a random variable x. The expected value or mean of this distribution is denoted as μ and its standard deviation as σ. For sample of size n the expected value of the distribution of the sample means is also μ. The standard deviation of the distribution of sample means is however σ/√n. So as the sample size increases the distribution of the sample means becomes more concentrated about the mean value.

Below are shown the histograms for samples of various sizes. In order to more clearly show the shape of the distribution the scale in the display is stretched as a function of sample size.

The random variable is uniformly distributed from -0.5 to +0.5; i.e.,

p(x) = 1 for -0.5≤x≤+0.5
p(x) = 0 for all other values of x

The value of μ for this distribution is 0. In the displays below the horizontal scale for the histogram for samples of size n is from -1/√n to +1/√n.

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