Laws of Fluctuations 735 



cients of the binomium, all further details seem 

 to depend upon them. In respect to the average 

 this is no doubt the case ; it is an empirical value 

 without need of any further discussion. The 

 more the number of the observations increases, 

 the more assured and the more correct is this 

 mean value, but generally it is the same for 

 smaller and for larger groups of observations. 



This however, is not the case with the ex- 

 tremes. It is quite evident that small groups 

 have a chance of containing neither of them. 

 The more the number of the observations in- 

 creases, the larger is the chance of extremes. As 

 a rule, and excluding exceptional cases, the ex- 

 treme deviations will increase in proportion to 

 the number of cases examined. In a hundred 

 thousand beans the smallest one and the largest 

 one may be expected to differ more widely from 

 one another than in a few hundred beans of the 

 same sample. Hence the conclusion that ex- 

 tremes are not a safe criterion for the discus- 

 sion of the curves, and not at all adequate for 

 calculations, which must be based upon more 

 definite values. 



A real standard is afforded by the steepness 

 of the slope. This may be unequal on the two 

 sides of one curve, and likewise it may differ for 

 different cases. This steepness is usually meas- 

 ured by means of a point on the half curve and 



