be or 0.150000 (see the second column of the table for 800 facts and 120 deaths). At first sight the medical observer would ap pear to be justified in asserting that under his method of treatment the mortality was at the rate of only 150,000 in 1,000,000 patients, or 15 per cent. But this assertion would be immediately met by the objection that the number of facts is not sufficient to justify this statement, that an average deduced from so small a number as 800 facts can only be re ceived as an approximation to the truth, and that it requires to be corrected by the aid of the figures in the table.
Accordingly, on referring to the column of possible errors corresponding to 800 cases and 120 deaths, we find that the error in ex cess and defect to which this number of facts is liable, amounts to 0.035707, which error must be added to and taken from the 0'150000, the result of actual observation. It follows, therefore, that the true result must be somewhere between the numbers 0.150000 added to 0.035707, or 0.185707, and dim. by 0.035707, or 0.114293, 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.
Uncorrected observation, therefore, would give, as the result of the treatment adopted, 13 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 sur la FO)re Typhoidc, has attempted to illustrate the treat ment of typhus fever, by minutely analysing 140 cases. The result was as follows : Number of deaths (nn) 52 Number of recoveries (n) 88 Total (2) 140.
The mortality in these cases was therefore or 0 37143 ; 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 . m n = V 2 / 2 ( 140) 52 . 88 — 0.11550.
3 This being the possible error in excess and defect, the true influence of the treat ment will be comprised between the following limits : 2 2.nz.n -+ • / + =0'37113 0'11550=0'48693 V and nz - 2 2.m.n=0'37143-0'11550=0'25593.
II 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 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.