Effects of External Conditions 107 
Since one of the primary objects of such investigations as the 
present is an inquiry into the effects of differences in the conditions 
of life upon supposedly homogeneous material, one of the principal 
points to be determined is the significance of any differences 
which may be discovered between the average values of a given 
character in two groups of individuals whose history has differed. 
For this purpose it is necessary to compare the probable error of 
the difference with the actually obtained difference between the two 
averages in question. The probable error of the difference 1s 
expressed by the formula: 
V/ (probable error of the first average)” + (probable error of the second average)” 
i. e, the square root of the sum of the squares of the probable 
errors of the two respective averages. Now the actually obtained 
difference is the most probably true difference and it is as likely 
to be too small as too large. Nevertheless the true difference 
may possibly equal 0, 1. e., be non-existent, in which case the ob- 
tained difference would be regarded as wholly “accidental.” 
From the table of the values of the “ probability integral” it may 
be calculated that the chances that a difference between two aver- 
ages is due to mere accident are: 
250 out of 1000 when difference between averages = 1 % probable error of the difference. 
156 out of 1000 when difference between averages = 1,5 X probable error of the difference. 
89 out of 1000 when difference between averages = 2 % probable error of the difference. 
46 out of 1000 when difference between averages = 2.5 X probable error of the difference. 
21 out of 1000 when difference between averages = 3 XX probable error of the difference. 
g out of 1000 when difference between averages = 3.5 % probable error of the difference. 
3 out of 1000 when difference between averages = 4 % probable error of the difference. 
than + out of 1000 when difference between averages = 4.5 X probable error of the difference, 
In proportion as the probability decreases that such a difference 
as been due to mere chance or accident (i. e., that it is the result 
of a multitude of independent causes having no relation to the 
conditions of the experiment), it is obvious that the probability 
increases that some constant modifying influence has been oper- 
ative in differentiating the two groups. It must be admitted 
however, that the probabilities here stated apply in full strictness 
