Thayer Watkins
Silicon Valley
& Tornado Alley

The Distribution of Sample Standard Deviation
(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 sum of the squares of a random variable which is uniformly distributed between -0.5 and +0.5. With larger n the distribution would be more concentrated so the horizontal scale is adjusted with sample size. Althought the random variable is distributed between -0.5 and +0.5 its square is distributed between 0 and 0.25 and the positive square root between 0 and +0.5.

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

As can be seen, as the sample size n gets larger the distribution of sample variance 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.

If the square root is taken of the mean value of the squares the distributions of the results are as is shown below:

The positive square root of the square of the random variable is distributed from 0 to 0.5. Although the distributions for larger sample size look generally like normal distributions they are transforms of normal distributions.

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