STATISTICS. 



807 



ations of the relative weight of the lungs and 

 body ; of wliich three observations, one was 

 made upon the body of an immature infant, 

 so that the subjects of the observations were 

 not strictly comparable. Though Ploucquet, 

 in this procedure, offended against two of the 

 most obvious statistical rules, his test con- 

 tinued to be treated with undeserved respect, 

 till comparatively recent investigations on a 

 larger scale had demonstrated the little re- 

 liance to be placed upon it. 



The most common attention to the ordinary 

 daily occurrences of life would suffice to 

 caution us against such errors as that into 

 which Ploucquet fell. Coincidences of the most 

 .startling character are constantly happening 

 to put us on our guard against them. One 

 which occurred to the writer of this article 

 deserves to be put on record. Two cases of 

 congenital absence of the larger pectoral 

 muscle on the same side of the body, oc- 

 curred, on the same day, among the out- 

 patients of the King's College Hospital. 

 This defect he has never happened to observe 

 within the wards of that Hospital or else- 

 where, either before or since. A similar 

 coincidence, though of a less striking cha- 

 racter, presented itself in the same institution 

 while the writer was noticing with some care 

 and interest the complexion and physiognomy 

 of patients suffering from pulmonary consump- 

 tion. His own previous experience, in con- 

 formity with the general opinion, had pointed 

 out the fair complexion as that of the great 

 majority of phthisical patients ; but the almost 

 exclusive occurrence for several days together 

 of the olive complexion, among patients 

 labouring under that disease, had almost led 

 him to discard his former opinion and that of 

 the best authorities, and to embrace one which, 

 as farther observation convinced him, would 

 have been erroneous. 



Games of chance are constantly furnishing 

 striking examples of these coincidences, in 

 the shape of what is familiarly known as a run 

 of good or ill-luck ; the same event, favourable 

 or unfavourable, occurring many times in suc- 

 cession, contrary not only to reasonable ex- 

 pectation, but to the results of unerring cal- 

 culation. On the other hand, the success of 

 the bank, with only a slight calculated chance 

 in its favour, but with a capital sufficiently 

 large to await the inevitable change in the run 

 of luck, vindicates the sufficiency of large 

 numbers of facts. The great annual fluctu- 

 ations, too, which take place in the balance of 

 the receipts and expenditure of assurance 

 offices, but the ultimate safety of their trans- 



of 



of in- 

 surances, serve to enforce the same truth. 



The sufficiency for all practical purposes 

 of large numbers of facts, may also be interred 

 from the remarkable uniformity observed to 

 take place in the annual summaries of events 

 brought about by the continued operation of 

 the same combination of causes. The annual 

 reports of the Registrar-General supply many 

 illustrations of this principle. The illustration 



actions, when extending over a long term 

 years and embracing a large number 



best suited to our present purpose, is one 

 drawn from an event removed, by the very 

 nature of the case, beyond the reach of ex- 

 ternal influences, or only very remotely and 

 indirectly amenable to them ; namely, the 

 proportion of male and female births in suc- 

 cessive years. In the eighth annual report 

 of the Registrar-General (p. Ixi.), a table is 

 given, in which the number of males and 

 females born, to every hundred living males 

 and females respectively, is recorded for the 

 seven years 1 839-45. If we substitute 

 100,000 for 100, the table will read thus. 



The largest excess of male over female 

 births, therefore, in these seven years is 295, 

 and the least 287, the average being 291 ; so 

 that the extreme fluctuation amounts to only 

 8 births in about 6500, or considerably less 

 than 1 in 800 ; while the excess or defect above 

 or below the average of 291 is only 4 births, 

 or less than 1 in 1600. If the causes which 

 determine the births of males and females re- 

 spectively could be assumed to be constant 

 and uniform, these fractional fluctuations 

 would express the divergences due to the 

 insufficiency of the number of observations 

 to express an absolutely true result. The 

 close approximation actually obtained must 

 be held to prove the sufficiency for every prac- 

 tical purpose of results based upon large 

 numbers of observations. 



Having thus shown, by two opposite ex- 

 amples, the total insufficiency of small num- 

 bers of facts, and the sufficiency, at least for 

 practical purposes, of large numbers of ob- 

 servations, it will be necessary to enter into a 

 more detailed examination of the relative 

 value of numbers of observations intermediate 

 between these two extremes. 



From what has been already stated, it must 

 be obvious that the degree of confidence to 

 be reposed in results based upon different 

 collections of facts must vary with the num- 

 ber of those facts ; and that, other things 

 being equal, the value of the results must 

 increase with every addition made to that 

 number. But it is only by actual observation, 

 or by mathematical calculations based upon 

 indisputable data, that the precise value of 

 any particular number of facts can be deter- 

 mined. Observation, indeed, is altogether 

 unequal to give more than a vague and general 

 idea of the relative values of small and large 

 collections of facts ; so that we must ulti- 

 mately resort to the mathematics both for 

 authoritative decisions and safe guides. As, 

 however, the large majority of mankind is 

 destitute of that mathematical knowledge and 



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