Statistical Method. 33 



3 and 4 are +22 and +17, with an average of +21 for all periods fol- 

 lowing the dose. 



Having obtained these differences with these particular data, we 

 should know what degree of reliability attaches to them. Since the 

 reaction times in table 2 form only a small series, we may reasonably 

 expect that if eye reactions were taken on many normal and alcohol 

 days the final resulting differences would not be exactly the same as 

 those obtained. We may ask: "Are the differences which we have 

 found due to chance or is it likely that a more extensive series or several 

 series of experiments would confirm this one by yielding differences 

 which, if not exactly equal in magnitude, would at least be the same in 

 direction?" To answer this question it is necessary to find (1) the 

 probable error of the difference (P. E. D ); (2) to calculate the ratio 



between the probable error of difference and the difference ( ) ; 



\P. E. D / 



and finally (3) from a table of the probability integrals to read off for 

 this particular ratio the probable correctness of the difference. 1 Sec- 

 tion in of the illustrative table 2 contains figures for the last two of 

 these expressions. Obviously a difference is considered reliable and 

 significant according to the relative size of its P. E. D . If in a particu- 

 lar instance p ^ = 1.0 (i. c, the probable error of difference is just 

 Jr. ±L. D 



as large as the difference) , the value of the probability integral for this 

 ratio is 0.50. 2 In other words, there are 50 chances out of 100 that the 

 difference will deviate from the observed value by an amount greater 

 than that value. In only one-half of these 50 chances, or in 25 cases, 

 will the change be a decrease (a difference with opposite sign) . There- 

 fore the probable correctness of this difference would be 100 — 25=75, 

 usually written 0.750. A probable correctness of 0.500 represents pure 

 chance, while 1.000 is certain cause. Usually values above 0.900 are 

 considered to indicate reliable, significant differences. The probable 

 correctness figures in table 2, section in, are all above 0.900, notwith- 

 standing that only 5 reactions were taken in each experimental period, 



1 The mean variation, as has been noted, expresses the degree of variability of the individual 

 reaction times from the average. Likewise every individual average also varies somewhat from 

 the theoretically true average which would result from several series of experiments. The ex- 

 pression ± — — ' — ', where M. V. represents the mean variation and n the number of obser- 



V n 

 vations (reaction times) is known as the probable error of the mean (P. E.m) and is a measure of 



this variability of the average. The probable error of difference, P. E.d = y/P- £. 2 mi +P- E?mi, 

 i. e., the square root of the sum of the squares of the probable errors of the means. 



2 Boring has recently contributed two articles which illustrate very clearly the method of 

 obtaining and the usefulness of the "probable correctness of the difference"; see "The number of 

 observations upon which a limen may be based," Am. Journ. Psychol., 1916, 27, p. 315, and "On 

 the computation of the probable correctness of differences," Am. Journ. Psychol., 1917, 28, 

 p. 454. 



