Page:Biometrika - Volume 6, Issue 1.djvu/10

Substituting this value of $D$ we get

$\mu _{2}={\frac {\sigma ^{2}}{2n}}\left(1-{\frac {1}{4n}}\right)$, $\mu _{3}={\frac {\sigma ^{3}{\sqrt {1-{\frac {3}{2n}}+{\frac {1}{16n^{2}}}}}}{2n^{2}}}$, $\mu _{4}={\frac {3\sigma ^{4}}{4n^{2}}}\left(1+{\frac {1}{2n}}-{\frac {1}{16n^{2}}}\right)$.

Consequently the value of the standard deviation of a standard deviation which we have found $\left({\frac {\sigma }{{\sqrt {2n}}{\sqrt {1-{\frac {1}{4n}}}}}}\right)$ becomes the same as that found for the normal curve by Professor Pearson ($\sigma /{\sqrt {2n}}$) when $n$ is large enough to neglect the $1/4n$ in comparison with $1$.

Neglecting terms of lower order than ${\frac {1}{n}}$ we find

$\beta _{1}={\frac {2n-3}{n(4n-3)}}$, $\beta _{2}=3\left(1-{\frac {1}{2n}}\right)\left(1+{\frac {1}{2n}}\right)$.

Consequently as $n$ increases $\beta _{2}$ very soon approaches the value $3$ of the normal curve, but $\beta _{1}$ vanishes more slowly, so that the curve remains slightly skew.

Diagram I. Frequency curve giving the distribution of Standard Deviations of samples of 10 taken from a normal population.

Equation $y={\frac {N}{7\cdot 5\cdot 3}}{\frac {10^{\frac {9}{2}}}{\sigma ^{9}}}{\sqrt {\frac {2}{\pi }}}x^{8}e^{-{\frac {10x^{2}}{2\sigma ^{2}}}}$.

Diagram I shows the theoretical distribution of the S.D. found from samples of 10.

$y={\frac {N\,10^{\frac {9}{2}}}{7\cdot 5\cdot 3}}{\sqrt {\frac {2}{\pi }}}{\frac {x^{8}}{\sigma ^{9}}}e^{-{\frac {10x^{2}}{2\sigma ^{2}}}}$.

Section V.

Some properties of the curve $y={\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}...{\binom {{\frac {4}{3}}\cdot {\frac {2}{\pi }}\;{\textrm {if}}\;n\;{\textrm {be}}\;{\textrm {even}}}{{\frac {5}{4}}\cdot {\frac {3}{2}}\;{\textrm {if}}\;n\;{\textrm {be}}\;{\textrm {odd}}}}(1+z^{2})^{-{\frac {n}{2}}}$.

Writing $z={\textrm {tan}}\,\theta$ the equation becomes $y={\frac {n-2}{n-3}}\cdot {\frac {n-4}{n-5}}\,{\textrm {...}}\;{\textrm {etc.}}\times {\textrm {cos}}^{n}\theta$, which affords an easy way of drawing the curve. Also $dz=d\theta /{\textrm {cos}}^{2}\theta$.