XIII. The law of the geometric mean

Suppose we have before us a large number of measurements. They may either be all approximations to the true value of a single unknown quantity, or may refer to the several members of a large class. The measurements will disagree among themselves, but on arranging them in order of size they show a tendency to cluster round some medium value. We are naturally inclined to infer that the true value of the unknown, or typical member of the class, is not far from this value. How to define and determine the appropriate medium in various classes of measurement becomes thus a natural object of inquiry. On examination we find that there is no strict and final criterion applicable to all cases.