Page:Popular Science Monthly Volume 71.djvu/180

174 not to exceed or to fall short of it. This, by construction, is the case of either quartile,

(New words and meanings—scheme of distribution of deviates, axis, normal, skew, quartile and probable error.)

In the fourth lesson it has to be explained that the curve of normal distribution is not the direct result of calculation, neither does the formula that expresses it lend itself so freely to further calculation, as that of frequency. Their shapes differ; the first is an ogive, the second (Fig. 7) is bell-shaped. In the curve of freqency the derivations are reckoned from the mean of all the variates, and not from the median. Mean and median are the same in normal curves, but may differ much in others. Either curve can be transformed into the other, as is best exemplified by using a polygon (Fig. 8) instead of the curve, consisting of a series of rectangles differing in height by the same amounts, but having widths respectively representative of the frequencies of 1, 3, 3, 1. (This is one of those known as a binomial series, whose genesis might be briefly explained.) If these rectangles are arrayed in order of their widths, side by side, they become the equivalents of the ogival curve of distribution. Now if each of these latter rectangles be slid parallel to itself up to either limit, their bases will overlap and they become equivalent to the bell-shaped curve of frequency with its base vertical.

The curve of frequency contains no easily perceived unit of variability like the quartile of the curve of distribution. It is therefore not suited for and was not used as a first illustration, but the formula that expresses it is by far the more suitable of the two for calculation. Its unit of variability is what is called the "standard deviation," whose genesis will admit of illustration. How the calculations are made for finding its value is beyond the reach of the present lessons. The calculated ordinates of the normal curve must be accepted by the learner much as the time of day by his watch, though he be ignorant of the principles of its construction. Much more beyond his reach are the formulas used to express quasi-normal and skew curves. They require a previous knowledge of rather advanced mathematics.

(New words and ideas—curve of frequency, standard deviation, mean, binomial series.)

The fifth and last lesson deals with the measurement of correlation, that is, with the closeness of the relation between any two systems whose variations are due partly to causes common to both, and partly to causes special to each. It applies to nearly every social relation, as to environment and health, social position and fertility, the kinship of parent to child, of uncle to nephew, etc. It may be mechanically illustrated by the movements of two pulleys with weights attached.