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

The skewness or asymmetry about the mean for a distribution can be measured in terms of the mean value of the cube of the deviations for its mean.

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 cube of x can then only have values between -0.125 and +0.125. Thus the probability density function for w=x3 is given by

#### P(w) = (1/3)w-2/3 for 0≤w≤0.125 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 cubes of a random variable which is uniformly distributed between -0.5 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 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. The mean value is zero.