Page:Biometrika - Volume 6, Issue 1.djvu/11

Rh Hence to find the area of the curve between any limits we must find

and by continuing the process the integral may be evaluated.

For example, if we wish to find the area between $$0$$ and $$\theta$$ for $$n=8$$ we have

and it will be noticed that for $$n=10$$ we shall merely have to add to this same expression the term $$\frac{1}{7}\cdot\frac{6}{5}\cdot\frac{4}{3}\cdot\frac{2}{\pi}\textrm{cos}^7\theta\,\textrm{sin}\theta$$.

The tables at the end of the paper give the area between $$-\infty$$ and $$z$$

This is the same as $$.5+$$the area between $$\theta=0$$, and $$\theta=\textrm{tan}^{-1}z$$, and as the whole area of the curve is equal to $$1$$, the tables give the probability that the mean of the sample does not differ by more than $$z$$ times the standard deviation of the sample from the mean of the population.

The whole area of the curve is equal to

and since all the parts between the limits vanish at both limits this reduces to $$1$$.

Similarly the second moment coefficient is equal to