Page:Biometrika - Volume 6, Issue 1.djvu/7

Rh Now

Summing for all values and dividing by the number of cases we get

where $$R_{u^2s^2}$$ is the correlation between $$u^2$$ and $$s^2$$.

Hence $$R_{u^2s^2}\sigma_{u^2}\sigma_{s^2}$$ or there is no correlation between $$u^2$$ and $$s^2$$.

To find the equation representing the frequency distribution of the means of samples of $$n$$ drawn from a normal population, the mean being expressed in terms of the standard deviation of the sample.

We have $$y=\frac{C}{\sigma^{n-1}}s^{n-2}e^{-\frac{ns^2}{2\sigma^2}}$$ as the equation representing the distribution of $$s$$, the standard deviation of a sample of $$n$$, when the samples are drawn from a normal population with standard deviation $$\sigma$$.

Now the means of these samples of $$n$$ are distributed according to the equation

and we have shown that there is no correlation between $$x$$, the distance of the mean of the sample, and $$s$$, the standard deviation of the sample.

Now let us suppose $$x$$ measured in terms of $$s$$, i.e. let us find the distribution of $$z=\frac{x}{s}$$.

If we have $$y_1=\phi(x)$$ and $$y_2=\psi(z)$$ as the equations representing the frequency of $$x$$ and of $$z$$ respectively, then