124 THE RATE OF GROWTH [ch. 



especially to know two important things. We want a mean value, 

 as a substitute for the true value* if there be such a thing; let us 

 use the arithmetic mean to begin with. About this mean the ob- 

 served values are grouped like a target hit by skilful or unskilful 

 shots ; we want some measure of their inaccuracy, some measure of 

 their spread, or scatter, or dispersion, and there are more ways than 

 one of measuring and of representing this. We do it visibly and 

 graphically every time we draw the curve (or polygon) of frequency ; 

 but we want a means of description or tabulation, in words or in 

 numbers. We find it, according to statistical mathematics, in the 

 so-called index of variability, or standard deviation (o), which merely 

 means the average deviation from the meanf. But we must take 

 some precautions in determining this average; for in the nature of 

 things these deviations err both by excess and defect, they are 

 partly positive and partly negative, and their mean value is the mean 

 of the variants themselves. Their squares, however, are all positive, 

 and the mean of these takes account of the magnitude of each 

 deviation with no risk of cancelling out the positive and negative 

 terms: but the "dimension " of this average of the squares is wrong. 

 The square root of this average of squares restores the correct 

 dimension, and the result is the useful index of variability, or of 

 deviation, which is called o-J. 



This standard deviation divides the area under the normal curve 

 nearly into equal halves, and nearly coincides with the point of 

 inflexion on either side; it is the simplest algebraic measure of 

 dispersion, as the mean is the simplest arithmetical measure of 

 position. When we divide this value by the mean, we get a figure 



* It is not always obvious what the "errors" are, nor what it is that they depart 

 or deviate from. We are apt to think of the arithmetic mean, and to leave it 

 at that. But were we to try to ascertain the ratio of circumference to diameter 

 by measuring pennies or cartwheels, our "errors" would be found grouped round 

 a mean value which no simple arithmetic could define. 



t a, the standard deviation, was chosen for its convenience in mathematical 

 calculation and formulation. It has no special biological significance; and a 

 simpler index, the "inter-quartile distance," has its advantages for the non- 

 mathematician, as we shall see presently. 



X That is to say : Square the deviation-from-the-mean of each class or ordinate 

 (^); multiply each by the number of instances (or "variates") in ^that class (/); 



divide by the total number (N) ; and take the square-root of the whole : a^= ~ "^ *' . 



I 



