226 E. C. MACDOWELL AND E. M. VICARI 



control averages, excepting retention for the males, and the second 

 half of training for the females; these two pairs of averages 

 give ratios very close to 1.000. 



Combining the sexes (primary averaging, disregarding sex) 

 and taking each strain separately, all the ratios are 'plus' except- 

 ing retention for strains A and C, and the second half of training 

 for strain A when the 'incompletes' are included with and without 

 the 'failures.' The highest of these 'minus' ratios is (— )1.090. 



When the strains as well as the sexes are averaged together, 

 'plus' ratios are given in every case. The differences and the 

 probable errors of the differences between the averages are 

 given in table 2. (The probable error of the difference is the 

 square root of the sum of the squares of the probable errors of 

 the two averages compared.) In a separate column are given 

 the quotients of the differences divided by their probable errors. 

 A quotient of three or more is usually regarded by statisticians 

 as certainly indicating a significant difference. In the first half 

 of training, omitting the first day, all training, and training and 

 retention the differences are more than three times their probable 

 errors and may be considered to be real or significant differences. 

 In the second half of training the 'completes' alone give a signifi- 

 cant difference, when the 'incompletes' and when the 'failures' 

 are included, the differences are 1.99 and 1.74 times their prob- 

 able errors; in retention the differences are 2.06 and 2.24 times 

 their probable errors. So besides finding many more 'plus' 

 than 'minus' ratios, the differences are shown to be statistically 

 significant when grouped to obtain large enough numbers to 

 calculate probable errors. 



The frequency distribution of the averages for each rat, includ- 

 ing 'failures,' from which the averages in the preceding tables 

 were obtained, are shown graphically in figure 2. For each group 

 of trials there is a pair of overlapping curves; the solid line repre- 

 sents the controls, the broken line the tests. The number of 

 rats in the test and control groups is so nearly equal that a 

 reduction to a percentage basis is not necessary. The graphs 

 show a clear tendency for the distribution of the tests to lie 

 further up the scale than the distributions of the controls; at 



