METHODS OF STATISTICAL ANALYSIS. 19 



make an enormously greater difference in the average weight of the 

 series than it will in the average pulse-rate, for body-weight is a far 

 more variable character than pulse-rate. The trustworthiness of a 

 constant based on a series of measurements is inversely proportional 

 to the variability of the individual measurements. On the other hand 

 it is reasonable to assume that the precision of a statistical constant 

 increases as the number of observations upon which it is based becomes 

 larger. Thus the average metabolism of 100 individuals is admittedly 

 more desirable as a basis for physiological generalization than an aver- 

 age based on 10 individuals; yet the trustworthiness of the constants 

 is not directly proportional to the number of observations upon which 

 they are based, but stands in the ratio of the square roots of these 

 numbers. Thus the probable error of an average based on 10,000 

 individuals would not be 100/10000 = 1/100 of that based on 100 

 individuals, but only VlOO/\/10000 = l/10. The practical conse- 

 quence of this relationship is that while precision increases with the 

 number of the observations, the increase in precision is not directly 

 proportional to the labor involved in the making of the measurements. 

 After a degree of precision which meets the practical requirements is 

 attained, further work may be regarded as lying beyond the limit of 

 diminishing returns. Of course the need of greater refinement may at 

 any time arise and demand the accumulation of a number of data 

 which for earlier work would have been considered superfluous. 



Details concerning the calculation of the probable errors a term 

 having an historical significance and not as appropriate as might be 

 found which can be obtained from text books on statistical methods, 

 need not detain us here. A few words are in order concerning the inter- 

 pretation of the probable error, the value appended with a plus and 

 minus sign to the various statistical constants. It is in reality a 

 measure of the variability of that constant which would be found if it 

 could be determined an infinitely large number of times upon random 

 samples of the same number of measurements and drawn from the same 

 population as that upon which the constant is based. It is a measure 

 of this variability of the statistical constant about its mean so chosen 

 that half of the values would lie inside and half of them outside the 

 limits of the probable error. Thus if the mean value of a character in 

 an infinitely large population were 86 and the probable error for samples 

 of 100 were ==5, 86=^=5 would indicate that if a large series of samples 

 of 100 individuals each were drawn at random from this population 

 half of these would show averages ranging from 81 to 91 while the 

 remaining 50 per cent would lie below 81 and above 91. 



The distribution of these means based on random samples of 100 

 individuals each would be an orderly one. Thus in the comparison 

 of two means it is possible for the statistician to estimate the chances 

 for (or against) their being based on identical material. Or, conversely, 



