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

The degree of flatness versus peakedness of a distibution can be measured by the mean value of the fourth power of deviations from the mean value.

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 fourth power of x can then only have values between 0 and 0.0625. Thus the probability density function for w=x4 is given by

#### P(w) = (1/4)w-3/4 for 0≤w≤0.0625 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 fourth powers 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 changed with sample size. Althought the random variable is distributed between -0.5 and +0.5 its fourth power is distributed between 0 and 0.0625.

Each time the display is refreshed a new batch of 2000 repetitions 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 reasonably close to a normal distribution even though the sample value can take on only positive values.