4
The Probable Error of a Mean
In a similar tedious way I find:
,
and
.
The law of formation of these moment coefficients appears to be a simple one, but I have not seen my way to a general proof.
If now be the th moment coefficient of about its mean, we have
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Hence
Consequently a curve of Professor Pearson’s type III. may be expected to fit the distribution of .
The equation referred to an origin at the zero end of the curve will be
,
where
,
and
.
Consequently the equation becomes
,
which will give the distribution of .
The area of this curve is (say).