[Illustration:  Fig. 1.]

[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 fractional parts are the middle (giving the median), the outside quarters (giving the upper and lower quartiles), and similarly the upper and lower octiles or deciles.  This is sufficient for most purposes.  It leaves only the outer eighths or tenths of the cases undescribed and undetermined, except so far as may be guessed by, the run of the intermediate portion of the curve, and it defines all of the intermediate portion with as close an, approximation as is needed for ordinary or statistical purposes.

Thus the heights of all but the outer tenths of the whole body of adult males of the English professional classes may be derived from the five following ordinates, measured in inches, of which the outer pair are deciles:—­

    67.2; 67.5; 68.8; 70.3; 71.4.

Many other instances will be found in the Report of the Anthropometric Committee of the British Association in 1881, pp. 245-257.

When we desire to compare any two large statistical groups, we may compare median with median, quartiles with quartiles, and octiles with octiles; or we may proceed on the method to be described in the next paragraph but one.