Page:Biometrika - Volume 6, Issue 1.djvu/3

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Samples of $$n$$ individuals are drawn out of a population distributed normally, to find an equation which shall represent the frequency of the standard deviations of these samples.

If $$s$$ be the standard deviation found from a sample $$x_1 x_2 ...x_n$$ (all these being measured from the mean of the population), then

Summing for all samples and dividing by the number of samples we get the mean value of $$s^2$$ which we will write $$\bar{s}^2$$.

where $$\mu_2$$ is the second moment coefficient in the original normal distribution of $$x$$: since $$x_1$$, $$x_2$$, etc., are not correlated and the distribution is normal, products involving odd powers of $$x_1$$ vanish on summing, so that $$\frac{2S(x_1x_2)}{n^2}$$ is equal to $$0$$.

If $$M'_R$$ represent the $$R$$th moment coefficient of the distribution of $$s^2$$ about the end of the range where $$s^2=0$$,

Again

Now $$S(x_1^4)$$ has $$n$$ terms but $$S(x_1^2x_2^2)$$ has $$\frac{1}{2}n(n-1)$$, hence summing for all samples and dividing by the number of samples we get

Now since the distribution of $$x$$ is normal, $$\mu_4=3\mu_2^2$$, hence