VALIDATION OF MEASURING INSTRUMENTS 171 
N the number of cases. The standard deviation or standard 
error of the difference in means is obtained by the following 
formula: 
(8) €="V Om, tox 
Here om and go, are the standard deviations of the two 
means. 
These formulas are not reliable where is less than 2 5. 
The differentiating value of the test is determined by com- 
paring the difference in means of the successes and failures 
with the standard error of this difference. If the difference 
is twice as great as its error, the chances are 43 in 44 that 
the difference is real and in the direction indicated; if it is 
three times as great, the chances are 740 in 741. This state- 
ment of probability is derived from the normal surface of 
error. If, for instance, the difference in means is three times 
as great as its standard error, then on one side of the normal 
surface of error at a distance from the mean equal to three 
times the standard deviation, the difference will become zero 
and beyond that point the values will be reversed; that is, the 
mean which was formerly the greater will now be the lesser. 
Since in the normal surface of error 1/ 741 of the measures 
occur beyond 3 at one side of the curve, we conclude that 
the chances are 1 in 741 that the difference is not real and 
not in the direction indicated, or 740 in 741 that it is real 
and in the direction indicated. A good example of the use 
of this method is to be found in Fernald, Hayes, and Dawley 
(51) and Dewey, Child, and Ruml (44) (see Table 8). 
For practical purposes the investigator may consider a 
test to be significant if the difference in means of the suc- 
cesses and failures is two or more times as great as its stand- 
ard error. 
A different formula is used when a group is compared with 
a sub-group; for example, when good, fair, and poor workers 
combined are compared as a whole with one of these three 
sub-groups. 
The formula for the standard deviation of the difference