METHODS OF INTEEPOLATION. 337 



if we were to add up the squares of the errors ia these columns, we shoukl 

 be giving much greater weight to some portions of the series than to 

 others, because the liability to error is much greater in some parts than 

 ia others, as is evident from an inspection of column (2), where the mean 

 errors diminish from age 20 to age 35, or thereabouts, and then increase 

 up to the end of the series, being more than ten times larger at age 75 

 than they are at age 35. We wish to test the applicability of the ad- 

 justment-formula used to all parts of the series alike; and to do this we 

 must put the errors on an equality by dividing the terms in column (4) 



bythosein column (2). Each resulting value of — will denote the amount 



of actual error for every unit of mean error. The squares of these values 

 are entered in column (5). Jfow, the arithmetical mean of the squares of 

 the units of mean error ought to be equal to the arithmetical mean of 

 the squares of the proportional actual errors. But the former quantity 

 will be unity, therefore the latter ought to be unity also; that is, the 



arithmetical mean of all the values of f — ) ought to be, approximately, 



unity. This is the proposed test of a good adjustment. 



To see how far it is satisfied in the present instance, we compute the 

 arithmetical mean of the seventy terms in column (5), and find that it is 

 only 0.90. But we cannot expect that such a result will be found exactly 

 equal to unity in any given case, for, the number of terms in the series 

 being limited, the actual distribution of errors will vary considerably 

 from that which theory assumes. What we do require is, that the dif- 

 ference from unity should not be greater than the probable error of the 

 arithmetical mean. This probable error can be roughly comjDuted in 

 the usual way. Subtracting the mean value 0.90 from each of the 

 terms in column (5), we find that the sum of the squares of all the remain- 

 ders is 164.7, and consequently the probable error sought is — 



0.6745 /J^^M_ = 12 

 V 69x70 



It thus appears that the value 0.90 does not differ from unity by more 



than its probable error; so that the series (3) satisfies the proposed test, 



and may be regarded as well adjusted. The mean value of l^\ 



might, however, 'be brought nearer to unity, if desired, by re-adjusting 

 series (3) with the same formula or some other one from Table III. 



And here we may observe that when repeated adjustments are made,, 

 the order in which the formulas are used is immaterial. For instance, 

 if a given series is adjusted by the five term formula of Table III, and 

 that result is adjusted again by the seven-term formula, the series thus 

 obtained is precisely the same as though we had used the seven: terni 

 formula first and then the five-term one. And the ratio of the probable 

 error of the final series to that of the original one is, iui both, cases^. 

 theoretically, 



22 g .229 X .0822 = .0188 



