Statlxtu'nl fuvtstif/atioim 339 



and It will be clear that we could not determine n — the number of inde- 

 pendent caiisp-i^rcmpH in ifeneral — ^witliout a knowledge of a and /*. for which 

 we require higher moments than the first two. If we suppose with Gallon 

 that " nature tosses," i.e. a = b = ^, then clearly a knowledge of iii/{q, — n») will 

 give s/n and so determine the number of elements, or contributory cause- 

 groups. 



Gallon obtains (p. 43) the following results: 



The source of the divergence is two-fold. First, there is no theoretical means 

 of discovering the quartiles in a binomial of discrete terms. Gallon deter- 

 mined them by drawing a freehand curve through the tops of the plotted 

 binomial blocks in order to reach a continuous ogive. This method is not 

 capable of great accuracy. Secondly, although the standard deviation of 

 a binomial is well known, the probable deviation (or quartile) will not 

 be equal to •67449 multiplied by the standard deviation unless n is fairly 

 large. Still the deviations seem too large to he wholly attributable to this 

 cause. 



I have enlarged on this iiuuier because it provides an iilu^sl^;lUull ot the 

 cases in which a standard deviation can be determined and a quartile cannot. 

 On the other hand there is more than an assumption of a = 6 in Galton's 

 method. If we are given any data, for example statures of a definite group, 

 there appears to be no reason why the zero of stature should correspond to the 

 start of the binomial; nature is more likely to take its additions and sub- 

 tractions fronj some definite value, and zero stature to be not t)nly a great 

 Improbability, but an impossibility. Further, nature's unit of addition or 

 subtraction will not be that of the measurement of stature and this introduces 

 another unknown. When we approjich the problem with all these quantities — 

 rt, n and both the unit of addition (or subtraction) and the centre about which 

 nature works — unknown, we can still solve the problem of fitting a binomial 

 to our data. Experience shows, however, that in the great majority of cases 

 the equations lead to imaginary values for our constants, or to such as 

 are uninterpretable (e.g. n negative) on the basis of a simple binomial. In 

 other words nature does not work on the b;isis of a finite number of 

 indepoident cause-groups, such as ai-e assumed in tln^ binomial frequency; 



43—2 



