Ill] 



THE CURVE OF ERROR 



133 



will the frequency curve be obviously skew, but the geometric mean^ 

 not the arithmetic, becomes the most probable value*. Now the 

 logarithm of the geometric mean of a series of numbers is the 

 arithmetic mean of their logarithms; and it follows that in such 

 cases the logarithms of the variants, and not the variants them- 

 selves, will tend to obey the Gaussian law and follow the normal 

 curve of frequency f. 



The Gaussian curve, and the standard deviation associated with 

 it, were (as we have seen) invented by a mathematician for the use 



21 



33 



36 



24 27 30 



Length in mm. 

 Fig. 25 A. Curve of frequency of a population of minnows. 



39 



of an astronomer, and their use in biology has its difficulties and 

 disadvantages. We may do much in a simpler way. Choosing a 

 random example, I take a catch of minnows, measured in 3 mm. 

 groups, as follows (Fig. 25 A): 



Size (mm.) 13-15 16^18 19-21 22-24 25-27 28^30 31-33 34-36 37-39 

 Number 1 22 52 67 114 257 177 41 2 



* See especially J. C. Kapteyn, Skew frequency curves in biology and statistics, 

 Rec. des Trav. Botan. N4erland., Groningen, xin, pp. 105-158, 1916. Also Axel 

 M. Hemmingsen, Statistical analysis of the differences in body-size of related species, 

 Danske Vidensk. Selsk. Medd. xcvm, pp. 125-160, 1934. 



t This often holds good. Wealth breeds wealth, hence the distribution of 

 wealth follows a skew curve; but logarithmically this curve becomes a normal 

 one. Weber's law, in physiology, is a well-known instance; on the thresholds 

 of sensations, effects are produced proportional to the magnitudes of those 

 thresholds, and the logs of the thresholds, and not the thresholds themselves, 

 are normaUy distributed. 



