STATISTICS. 



813 



Table of the Limits of Variation compatible 

 witb an equal Chance relative to two observed 

 Events.* 



Provision having been thus made by the 

 two preceding tables for testing the suffi- 

 ciency of average results based upon different 

 numbers of facts relating to two alternatives, 

 and determining the possible variation or 

 limits of error to which the numbers of facts 

 in question, considered simply as facts, with- 

 out regard to their peculiar nature, are liable, 

 provision still remains to be made for testing 

 in like manner the value of those averages, 

 which belong especially to the domains of 

 ph\ siology and hygiene, namely, the average 

 number of the pulse and respiration, the 

 average age at death of different classes of 

 the community, &c. In the absence of tables 

 specially adapted to this purpose, it must 



found in the work of Gavarret, so often quoted, 

 from p. 143. onwards, and in the notes at p. 286. 

 et seq. 



* The formula employed in the construction of 

 this table is 



0-50 



Avhcre c, as before, represents the total of the ob- 

 served facts. The table will be found at greater 

 length at p. 230 of Gavarret's work on Medical 

 Statistics. 



suffice to state, in general terms, that the 

 averages derived from a given number of facts 

 are not to be regarded as strict expressions of 

 the truth, but as approximations more or less 

 remote, as the number of facts is less or more 

 considerable. 



But a very important question here arises : 

 To what extent, and under what restric- 

 tions, do calculations based on mathematical 

 formulae and derived from abstract reasoning, 

 admit of application to the results of actual 

 observation ? Conceding, as we may safely 

 do, the soundness of the formulae, there is 

 yet great room to doubt the propriety of their 

 application to the average results of observa- 

 tion. For if we suppose a mathematical 

 formula to be applied successively to a long 

 series of averages derived from the same 

 number of facts, it must obviously administer 

 a similar correction to those averages which 

 happen to coincide with the true average, and 

 to those which lie at the two extremes. This 

 consideration is sufficient in itself to condemn 

 the use of mathematical formulas, except as 

 a means of exhibiting in a striking light the 

 possible error attaching to a small number of 

 facts, considered abstractedly as facts. 



From the foregoing considerations, then, it 

 would seem to follow, that although averages 

 derived from small numbers of facts are sub- 

 ject to a considerable amount of possible 

 error, there is always such a probability of their 

 coinciding with, or not differing widely from, 

 the true averages, as to justify their employ- 

 ment as standards of comparison and data for 

 reasoning. At the same time it must be con- 

 ceded, that averages derived from small num- 

 bers of facts stand in need of a confirmation 

 which averages drawn from larger numbers of 

 facts do not require, and that in using the 

 former we are bound to speak with a reserve 

 proportioned to the scantiness of our ma- 

 terials. 



Of extreme values derived from observation. 

 As averages founded upon large numbers 

 of facts are numerical expressions of true pro- 

 babilities, so extreme values are expressions, 

 in the same precise language, of possibilities. 

 Both orders of facts have their scientific and 

 practical applications ; but those applications 

 which belong to the extreme values have been 

 less attended to than those which pertain to 

 averages. 



One obvious use of extreme values is to 

 confirm and strengthen the conclusions drawn 

 from averages. Thus, if we wish to ascertain 

 the relative duration of life of two classes of 

 persons, we may make use of the gredtest 

 age attained by either class in confirmation of 

 the mean of all the observations ; and the 

 coincidence of the one with the other will 

 give increased confidence in the general result. 

 Another important use of extreme values is 

 as a test of numerical theories. Two apt il- 

 lustrations of this application of figures are 

 afforded by that practical science which deals 

 most largely in possibilities Forensic Medi- 

 cine. M. Orfila, in his " Trait e des Exhuma- 

 tions," states that it is possible to determine 



