[Illustration: Fig. 2.]

The mathematical conception of the curve is best expressed in Fig. 2,
where PQ represents any given deviation from the average value, and
the ratio of PO to AB represents the relative probability of its
occurrence. The equation to the curve and a discussion of its
properties will be found in the _Proceedings of the Royal Society_,
No. 198, 1879, by Dr. M'Alister. The title of the paper is the
"Law of the Geometric Mean," and it follows one by myself on
"The Geometric Mean in Vital and Social Statistics."

We can lay down the ogive of any quality, physical or mental,
whenever we are capable of judging which of any two members of the
group we are engaged upon has the larger amount of that quality. I
have called this the method of statistics by intercomparison. There
is no bodily or mental attribute in any race of individuals that can
be so dealt with, whether our judgment in comparing them be guided
by common-sense observation or by actual measurement, which cannot
be gripped and consolidated into an ogive with a smooth outline, and
thenceforward be treated in discussion as a single object.

It is easy to describe any given ogive which has been based upon
measurements, so that it may be drawn from the description with
approximate truth. Divide AB into a convenient number of fractional
parts, and record the height of the ordinates at those parts. In
reproducing the ogive from these data, draw a base line of any
convenient length, divide it in the same number of fractional parts,
erect ordinates of the stated lengths at those parts, connect their
tops with a flowing line, and the thing is done. The most convenient