Page:Biometrika - Volume 6, Issue 1.djvu/12

Hence the standard deviation of the curve is $1/{\sqrt {n-3}}$. The fourth moment coefficient is equal to

${\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}\times \int _{-{\frac {\pi }{2}}}^{+{\frac {\pi }{2}}}{\textrm {cos}}^{n-2}\theta \,{\textrm {tan}}^{4}\theta d\theta$
$={\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}{\textrm {...}}\;{\textrm {etc.}}\times \int _{-{\frac {\pi }{2}}}^{+{\frac {\pi }{2}}}({\textrm {cos}}^{n-6}\theta -2{\textrm {cos}}^{n-4}\theta +{\textrm {cos}}^{n-2}\theta )d\theta$
$={\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}-{\frac {2(n-2)}{n-3}}+1={\frac {3}{(n-3)(n-5)}}$.

The odd moments are of course zero as the curve is symmetrical, so

$\beta _{1}=0$, $\beta _{2}={\frac {3(n-3)}{n-5}}=3+{\frac {2}{n-5}}$.

Hence as $n$ increases the curve approaches the normal curve whose standard is $1/{\sqrt {n-3}}$.

$\beta _{2}$ however is always greater than $3$, indicating that large deviations are more common than in the normal curve.

Diagram II. Solid curve $y={\frac {N}{S}}\times {\frac {8}{7}}\cdot {\frac {6}{5}}\cdot {\frac {4}{3}}\cdot {\frac {2}{\pi }}{\textrm {cos}}^{10}\theta$, $x/s={\textrm {tan}}\theta$.

Broken live curve $y={\frac {{\sqrt {7}}\cdot N}{{\sqrt {2\pi }}\cdot s}}e^{-{\frac {7x^{2}}{2s^{2}}}}$, the normal curve with the same S.D.

Distance of mean from mean of population

I have tabled the area for the normal curve with standard deviation $1/{\sqrt {7}}$ so as to compare with my curve for $n=10$^[1]. It will be seen that odds laid according to either table would not seriously differ till we reach $z=.8$, where the odds are about 50 to 1 that the mean is within that limit: beyond that the normal curve gives a false feeling of security, for example, according to the normal curve it is 99,986 to 14 (say 7000 to 1) that the mean of the population lies between $-\infty$ and $+1.3s$ whereas the real odds are only 99,819 to 181 (about 550 to 1).

1. See p. 19.