Page:Biometrika - Volume 6, Issue 1.djvu/5

Rh The first moment coefficient about the end of the range will therefore be

The first part vanishes at each limit and the second is equal to

and we see that the higher moment coefficients will be formed by multiplying successively by $$\frac{n+1}{n}\mu_2$$, $$\frac{n+3}{n}\mu_2$$, etc., just as appeared to be the law of formation of $$M'_2$$, $$M'_3$$, $$M'_4$$, etc.

Hence it is probable that the curve found represents the theoretical distribution of $$s^2$$; so that although we have no actual proof we shall assume it to do so in what follows.

The distribution of $$s$$ may be found from this, since the frequency of $$s$$ is equal to that of $$s^2$$ and all that we must do is to compress the base line suitably.

Now if

and

then

or

Hence

is the distribution of $$s$$.

This reduces to

Hence $$y=Ax^{n-2}e^{-\frac{nx^2}{2\sigma^2}}$$ will give the frequency distribution of standard deviations of samples of $$n$$, taken out of a population distributed normally with standard deviation $$\sigma$$. The constant $$A$$ may be found by equating the area of the curve as follows:—

Then

since the first part vanishes at both limits.