812 



STATISTICS. 



So that instead of asserting, as we should 

 seem justified in doing, that the mortality 

 under the influence of the treatment adopted 

 amounted to 15 per cent., we could only 

 claim a mortality comprised between the 

 numbers 



185,707 and 114,293 in 1,000,000 cases: 

 or approximatively between the numbers, 19 

 and 11 per cent. 



Unconnected observation, therefore, would 

 give, as the result of the treatment adopted, 

 15 per cent., while corrected observation 

 would give some number between 19 and 11 

 per cent. 



The application of the formula given in the 

 note to an actual case will be more instruc- 

 tive than an imaginary example. 



M. Louis, in his Recherches sitr In Ficvre 

 Typho'ide, has attempted to illustrate the treat- 

 ment of typhus fever, by minutely analysing 

 140 cases. The result was as follows : 

 Number of deaths (m ) 52 

 Number of recoveries (n) 88 

 Total - O) 140. 



The mortality in these cases was therefore 

 T \%, or 037143 ; and if we were to take this 

 mortality as the strict expression of the re- 

 sults of the treatment adopted, we should 

 shape our proposition as follows : The mor- 

 tality of typhus fever, under the treatment 

 adopted by M. Louis, amounted to 



37,143 deaths in 100,000 patients ; 

 or, in round numbers, 



37 deaths in 100 patients 



If, now, we proceed by means of the formula 

 referred to, to determine the possible error at- 

 taching to this proposition (i. e. to the num- 

 ber of facts upon which it is made to rest), 

 we find it to be equal to 

 2 / 2 . m . n 2 / 2 . 52 . 88 



{/ ~^~ = \/ 7T5o>r = 0-H550. 



This being the possible error in excess 

 and defect, the true influence of the treat- 

 ment will be comprised between the following 

 limits : 



and 



=0-37143-0- 11550=0-25593. 



Thus all that we really learn from this re- 

 cord of experience is, that, under the treat- 

 ment adopted, the number of deaths may vary 

 between 



48,693 and 25,593 in 100,000 patients, 

 or approximatively between 49 and 26 in 100 

 patients. 



In other words, if we were to employ the 

 same mode of treatment in a great number 

 of cases of typhus fever, we might lose any 

 number between about a fourth and a half of 

 our patients.* 



The same formula is equally applicable to 

 the solution of doubtful questions relative to 

 the results of two or more series of facts which 



* Gavarret, Principes G&ieraux de Statistique 

 Me'dicale, p. 284. 



we are desirous of comparing. It may happen 

 that the difference between the average result 

 of one series of facts and that of a second 

 series, is so inconsiderable, as to leave us in 

 doubt whether it may not be explained by a 

 reference to the limits of error to which the 

 number of facts in either return is liable.* 



It often happens, that the average results 

 of two series of observations relating to two 

 alternative events (such as the events of 

 death or recovery in particular diseases, the 

 birth of a male or female child, &c.) approxi- 

 mate so closely, that we are at a loss whether 

 to attribute the slight difference existing be- 

 tween the two averages to coincidence, or to 

 the operation of certain efficient causes. If 

 the number of observed facts be small, the 

 difference between the averages derived from 

 the two series of facts may be so slight as to 

 fall short of the difference between the limits 

 of error in excess or defect. The same re- 

 sult may also happen with any number of 

 facts, however considerable. In order to 

 solve the doubts which necessarily attach 

 to such close approximations, a mathematical 

 formula has been brought into requisition, 

 and employed in the formation of tables ap- 

 plicable to this purpose. Such a table is 

 subjoined. The mode of applying it will be 

 presently explained. 



The use of this table will be understood 

 from the following example : In the six 

 years 1839-44 there occurred in England, on 

 the average of those years, 515,478 births, of 

 which 264,245 were males, and 251,233 fe- 

 males. As the difference between these two 

 numbers is not very considerable, a question 

 might arise, whether that difference is not 

 compatible with the error in excess to which 

 half a million of facts is liable. The use of 

 the formula on which the foregoing table is 

 founded, would at once clear up this doubt. 



264,245 male births in a total of 515,478, 

 is equal to 



512,904 male births in 1,000,000 births. 

 But, on the supposition that the male and 

 female births are really equal in number, we 

 have the limits of variation equal to 500,000 



and 500,000 



515478 



-v 



515478, 



or 500,0004-000,624 and 500,000000,624, 

 or a maximum of 500,624, and a minimum of 

 499,376. The difference by the formula is, 

 therefore, 1248 in the million, while the ob- 

 served difference between the highest and 

 lowest number of male births occurring in the 

 six years 1839-44, is 514,809511,781, or 

 3,028. The inference, therefore, is irresist- 

 ible, that the excess of male births is due to 

 some efficient cause or causes, and that it is 

 not merely an error to which the number of 

 half a million of facts is inevitably exposed.f 



* For illustrations of this application of the fore- 

 going table, and of the formula fromwhich the figures 

 are calculated, see Gavarret's Statistique Me'dicale, 

 p. 80. et seq., and notes, p. 274. 



f Several applications of this table and of the 

 formula fromVhich the figures are derived, will be 



