810 



STATISTICS. 



relating to the duration of life of the aristo- 

 cracy, and the second on the facts relating to 

 the duration of life of the combined upper 

 and middle classes. For the sake of per- 

 spicuity, the average of all the facts in either 

 table is distinguished by a larger type. 



These tables speak for themselves. In the 

 first table, for instance, the small number of 

 25 facts is seen to yield the same average as 

 the total of 1800 facts in no less than 9 in- 

 stances, or one eighth of the whole number ; 

 while in 26 out of 72 instances, or more than 

 one third, the average of 25 facts exceeds or 

 falls short of the average of 1800 facts by 

 only a single unit. In like manner, it ap- 

 pears from the second table, that in 20 cases 

 out of 128, or little less than one sixth, the 

 average of 50 facts coincides with that of 

 G400^facts; and that in 42 out of 128, or 

 nearly one third, it differs from it only by a 

 single unit. Without entering into a minute 

 examination of the other columns of the two 

 tables, it will suffice to state that the proba- 

 bility in favour of an average of a given num- 

 ber of observations coinciding with the true 

 average increases with the number of observ- 

 ations ; so that we are again brought back to 

 the expediency of collecting large numbers 

 of observations wherever it is practicable so 

 to do. In using small numbers of facts to 

 establish data for reasoning or standards of 

 comparison, we are bound to speak with dif- 

 fidence of their sufficiency, and we ought to 

 regard them rather in the light of probabilities 

 requiring to be strengthened by other pro- 

 babilities, as weak arguments require to be 

 supported by additional reasons, than as, in 

 themselves, worthy of great reliance. Ac- 

 cording to this view of the case, we are not 

 precluded from the use of averages drawn 

 from small numbers of facts. The employ- 

 ment of such averages with this proviso is an 

 absolute scientific necessity ; for in many in- 

 stances we are prevented by causes too nu- 

 merous to specify from bringing together facts 

 by the hundred or the thousand, and yet, 

 were we to reject the smaller numbers as in- 

 admissible, we should be thrown back upon 

 the still more loose and less trustworthy 

 general statements from which it is the pro- 

 vince of statistics to rescue us. 



An examination of the two foregoing tables, 

 as well as of those which display the extreme 

 variations between the averages derived from 

 the same numbers of facts, will serve to prove 

 the hopelessness of any attempt to establish 

 by observation rules for measuring the rela- 

 tive value of averages derived from different 

 numbers of facts. It must be equally evident 

 that no deductions drawn from observation 

 can enable us to state the actual liability to 

 error of any given number of facts, considered 

 as facts, without reference to their peculiar 

 nature. To determine this liability to error 

 belongs solely to the mathematics. 



If, on the one hand, observation is unable 

 to supply us with the means of testing the 

 true liability to error of conclusions based on 



a given number of facts, considered as facts, 

 without reference to their peculiar nature, it 

 must be evident, on the other hand, that ma- 

 thematical formulae deduced from abstract 

 reasoning can only supply us with the means 

 of measuring the value of a given number of 

 facts in this their abstract relation, without 

 taking into account the varying quality of the 

 facts themselves. But as it is of the utmost 

 importance to be able to test the abstract 

 sufficiency of a given number of facts to 

 establish a principle or to supply a sound 

 standard of comparison, it will be necessary 

 to enter at some length into this part of our 

 subject. 



The facts already adduced, must have 

 abundantly shown that the limits of deviation 

 from a true average result are wider or nar- 

 rower as the number of facts from which the 

 average is drawn is smaller or greater. Many 

 eminent mathematicians, and M. Poisson 

 among the number, have laboured to convert 

 this general principle into an exact numerical 

 expression or formula, applicable as a test of 

 the true value of larger or smaller collections 

 of facts, and as an exact measure of the 

 limits of variation. M. Gavarret, in his work 

 on Medical Statistics, contends successfully 

 for the introduction of these formulas into 

 the service of the medical man ; and adopting 

 the sentiment of Laplace, " Le systeme tout 

 entier des connaissances humaines se rattache 

 a la theorie des probabilites," he insists that 

 medical statistics, or, as we prefer to term 

 it, the Numerical Method, applied to medicine, 

 is nothing more nor less than a special appli- 

 cation of the Calculus of Probabilities, and 

 the Theory of large Numbers ; and that as 

 such it is the most indispensable complement 

 of the experimental method. In other words, 

 he deems it incumbent on the medical man to 

 apply to his numerical results the corrections 

 supplied by the formulae of the pure mathe- 

 matics ; and before he concludes that any 

 number actually obtained by observation is a 

 true representative of a fact or law, to deter- 

 mine whether that number may not be com- 

 prised within the limits of possible variation. 

 M. Gavarret illustrates the necessity of this 

 precaution by applying his mathematical for- 

 mulae to a great variety of results based upon 

 observation ; but he especially insists upon 

 bringing the alleged efficacy of certain modes 

 of treatment to this searching test. The most 

 convenient course to adopt, in reference to 

 these formulae, will be to present the calcula- 

 tions based upon them in tabular forms, and 

 then to apply these calculations to one or 

 two striking examples. 



Tiie following table presents at one view 

 the possible errors corresponding to average 

 mortalities deduced from different numbers 

 of observations. It is obvious that the table 

 is equally applicable to other contingencies 

 of the same kind, where one of two events 

 is possible in every instance. The mode of 

 using it will be presently explained and illus- 

 trated. 



