Page:Biometrika - Volume 6, Issue 1.djvu/8

8 Hence

is the equation representing the distribution of $$z$$ for samples of $$n$$ with standard deviation $$s$$.

Now the chance that $$s$$ lies between $$s$$ and $$s+ds$$ is:

which represents the $$N$$ in the above equation.

Hence the distribution of $$z$$ due to values of $$s$$ which lie between $$s$$ and $$s+ds$$ is

and summing for all values of $$s$$ we have as an equation giving the distribution of $$z$$

By what we have already proved this reduces to

and to

Since this equation is independent of $$\sigma$$ it will give the distribution of the distance of the mean of a sample from the mean of the population expressed in terms of the standard deviation of the sample for any normal population.

By a similar method to that adopted for finding the constant we may find the mean and moments: thus the mean is at $$\frac{I_{n-1}}{I_{n-2}}$$, which is equal to

or