San José State University
Department of Economics
Thayer Watkins
Silicon Valley
& Tornado Alley

The Distribution of Sample Variance
(when the mean value of the
random variable is known)
as a Function of Sample Size

Suppose the probability density distribution for x is

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

The square of x can then only have values between 0 and 0.25. Thus the probability density function for w=x2 is given by

P(w) = w-1/2 for 0≤w≤0.25
P(w) = 0 for all other values of w

Below are shown the histograms for 2000 repetitions of taking samples of n random variables and computing the mean value of the squares of a random variable which is uniformly distributed between -0.5 and +0.5. The sum is normalized by dividing by the square root of the sample size n. This keeps the dispersion of the distribution constant. Otherwise with larger n the distribution would be more spread out. Althought the random variable is distributed between -0.5 and +0.5 its square is distributed between 0 and 0.25.

Each time the display is refreshed a new set of 2000 repetions of the samples is created.

As can be seen, as the sample size n gets larger the distribution more closely approximates the shape of the normal distribution.

Although the distribution for n=1 is decidedly non-normal, for n=16 the distribution looks quite close to a normal distribution even though the sample value can take on only positive values.

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