THE SPECIFICATION OF CONTINUOUS VARIATION 



such points, one on each side, but as the curve is symmetrical they 

 must lie at equal distances from /x. The estimate, s, of o- obtained 

 from any set of data is called the standard deviation and is found as 



S(x-i)2 



— I ^ ' 



the notation being as before. It is, however, more convenient for 

 purposes o£ calculation to use the algebraically identical formula 



n — I 





where S^{x) stand for the square of the sum of all the x's. 



It will be observed that s is found as the square root of s^, which 

 is generally called the variance, and is denoted by V. The variance 

 is in fact of more use than the standard deviation in genetical work, 

 because when variation is measured by the variance, the con- 

 tributions made to it by independent causes of variation are additive. 

 This obviously caimot be equally true when the variation is measured 

 by the standard deviation. We shall, therefore, regard the normal 

 curve as most conveniently specified by fx and a"; or rather by the 

 estimates of these quantities, x and V, as obtained from a set of 

 actual data. 



Not all frequency distributions of this general shape, with the 

 central values the most common, are normal curves. Some are 

 asymmetrical. And even some symmetrical curves do not conform 

 to the normal type. Such curves require further parameters, esti- 

 mated from the values of S (x— x)^ and S (x— x)*,for their full speci- 

 fication; but even so, the mean and variance still retain their crucial 

 importance for our understanding of these distributions. As we shall 

 see, much can be learned of the genetical situation underlying con- 

 tinuous variation by the analysis of means and variances, and this 

 still remains true even where we know the frequency distribution 

 to depart from strict normality. 



There is one further biometrical quantity which is of importance 

 for the genetical study of continuous variation, since it helps to tell 

 us directly the extent to which continuous differences are hereditary. 

 Where we have simultaneous observations on pairs of relatives, say 



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