STATISTICS. 



811 



Table of the possible Errors corresponding to Average Mortalities deduced from different Numbers 



of Observations.* 



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 : 

 1 20 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, 



y2 .m.n 

 y*3- 



in which TO represents the number of times that an 

 event A has happened, n the number of times that 

 an event u has happened, and ju. the total number 



of events : so that m + n 



- the average fre- 



quency 



5 



* 



the lim 



of the events TO,as obtained by observation ; and 



TO_ + /2 . TO .n an ,i A. _ /2 . TO . n 

 ft 2V f* 3 m 2 V A* 3 



the limits within which the true average,as corrected 



by the formula, lies. 



be |4g, 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 150000 dim. by 0-035707, or 0-114293. 



