CONTINUOUS VARIATION 



is, or how short a short one is, provided we can classify into the tall 

 and short categories. Scaling is, however, of obvious importance 

 in the analysis of continuous variation, since the scale will, in part, 

 determine the shape and properties of the distribution which are 

 all we have to base the genetical conclusions upon. We shall return 

 to it later. 



As we have seen, the type of grouped frequency distribution of 

 Fig. 14 is to be regarded as an approximation to the underlying 

 continuous distribution which we may seek to specify algebraically 

 in terms of a few parameters. Where the distribution is symmetrical 

 and the frequencies fall off on each side of the centre value, as in the 

 figure, this ideal curve can be, and indeed usually is, taken as an 

 example of the Gaussian or Normal Curve, whose general formula is 



I 



(x-M.)2 



df-:^^e 2a^ dx 



aV2TT 



where df is the frequency of individuals falling within the infini- 

 tesimal range, dx, of heights. Apart from f, the frequency, and x the 

 heights, four quantities appear in this formula 77, e, a and yu,. The 

 first two of these, 77 and e, are familiar mathematical constants, the 

 ratio of the circumference of a circle to its diameter and the natural 

 base of logarithms respectively. These will be the same for all normal 

 curves. The other two, a and ju,, are the parameters which specify 

 the particular normal curve in question, and they must be found for 

 each distribution individually. They are the constants by means of 

 which we can specify the variation of the character in which we are 

 interested. 



The way in which ju, and a specify the curve can be seen from 

 Fig. 14. /M is the value of the character at the centre of the dis- 

 tribution and is estimated as the mean, or average, of the values of 

 the character in all the individuals concerned. Thus where there 

 are n individuals whose height has been measured, ^i is estimated 



S(x) 

 by X = ' S standing for summation of the values of x from all 



individuals. yL, or its estimate x, fixes the position of the curve of 

 distribution. 



The second parameter, a, is the distance of the point of maximum 

 slope of the curve from the mean, /u,. There are, of course, two 



58 



