The second moment about the end of the range is
.
The third moment about the end of the range is equal to
the mean. |
The fourth moment about the end of the range is equal to
.
If we write the distance of the mean from the end of the range and the moments about the end of the range , , etc.
then
, , , .
From this we get the moments about the mean
, | |
, | |
. |
It is of interest to find out what these become when is large.
In order to do this we must find out what is the value of .
Now Wallis’s expression for derived from the infinite product value of sin is
.
If we assume a quantity which we may add to the in order to make the expression approximate more rapidly to the truth, it is easy to show that etc. and we get
.[1]
From this we find that whether be even or odd approximates to when is large.
- ↑ This expression will be found to give a much closer approximation to than Wallis’s.