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.
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 formula; deduced from abstract reasoning can only supply us with the means of measuring the value of a given number of filets 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 formulm 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 formula, 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.
The 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.
The use of this table will be best explained by an example. Let us suppose that a me dical man, having, for a long time, adopted a particular course of treatment in a certain malady, has arrived at the following results : 120 deaths, 680 recoveries, 800 cases.
The average mortality in this case would * This table is an abbreviation of one given at p.142. of Gavarret's work, but with additional cal culations based on the same formula, for the numbers from 25 to 200 inclusive. The formula from which the figures in the column of possible errors have been calculated is, . . n 2 Ks— in which m represents the number of times that an event A has happene0, n the number of times that an event B has happened, and ,u the total number of events : so that 7n+77=-/A; the average quency of the events m,as obtained by observation ; and m m n m n 2 K3 m 2 P3 the limits within which the true average,as corrected by the formula, lies.