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

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

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

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.



I have tabled the area for the normal curve with standard deviation $$1/\sqrt{7}$$ so as to compare with my curve for $$n=10$$. 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).