462 MR ROBERT WORTHINGTON. 



Eegarding now the frequency curve from the biological point 

 of view, we have simply a translation of terms. The elementary 

 errors whose concurrence determine the gross error become the 

 numerous contributory causes leading to variation from the mean. 

 The ordinate PN of any point P on the curve represents the 

 frequency of organs differing by NM from the mean. And the 

 'prohable error may now be called the 2>'>'ohable deviation. 



Frequency Constants. — The frequency curve is determined by 

 the values of certain constants which we shall now proceed to 

 define. 



The Mean is the abscissa of the ' centre of gravity ' of the area 

 enclosed by the frequency curve. 



The Mode is the abscissa of the ordinate of maximum frequency. 

 In the normal curve it will be observed that the mean and the 

 mode are identical. This is not the case, however, when the 

 frequency is asymmetrical. 



The Standard Deviation (o-) or ' index of variability ' is a 

 constant determining the slope of the curve. 



^sum of [(deviation of each group from the mean)"-^ 

 X frequency of group] 



a ■ 



number of organs 



Ci" 



'^{x'y) 

 N 



where x, y, are the co-ordinates of any point on the curve, and 

 N = total number of organs, 

 = area of curve, 

 o- is a quantity of the same order as the 'probable deviation' 

 mentioned above (prob. deviation = 0-6745 cr). 



The determination of the above constants from a frequency 

 polygon of actual measurements is only an approximation to 

 their true values. The closeness of the approximation to the 

 truth is called the probable error of the determination. 



The probable errors of frequency constants are discussed by 

 Professor Pearson and Mr L. N. G. Filon in "Mathematical 

 Contributions to the Theory of Evolution," No. IV., in the 

 Phil. Trans. Ptepeating our definition of the Mean in the 



abbreviated form 



^{x.y) ■ ■ 



