Page:Biometrika - Volume 6, Issue 1.djvu/9

Rh The second moment about the end of the range is

The third moment about the end of the range is equal to

The fourth moment about the end of the range is equal to

If we write the distance of the mean from the end of the range $$\frac{D\sigma}{\sqrt{n}}$$ and the moments about the end of the range $$v_1$$, $$v_2$$, etc.

then

From this we get the moments about the mean

It is of interest to find out what these become when $$n$$ is large.

In order to do this we must find out what is the value of $$D$$.

Now Wallis’s expression for $$\pi$$ derived from the infinite product value of sin $$x$$ is

If we assume a quantity $$\theta\left(=a_0+\frac{a_1}{n}+\textrm{etc.}\right)$$ which we may add to the $$2n+1$$ in order to make the expression approximate more rapidly to the truth, it is easy to show that $$\theta=-\frac{1}{2}+\frac{1}{16n}-$$etc. and we get

From this we find that whether $$n$$ be even or odd $$D^2$$ approximates to $$n-\frac{3}{2}+\frac{1}{8n}$$ when $$n$$ is large.