1919] Frost: Mutation in Matthiola 151 



such deviations lie in the opposite direction and represent positive 

 differences still greater than the sample difference. In other words, 

 if the implications of a sample difference are to be given fnll weight, 

 this difference must be considered the most prohable value of the 

 theoretical "true" difference between two assumed distinct statistical 

 populations. In the present case we wish to know the probability that 

 the "true" or theoretical-population means differ in the same sense 

 as the observed sample means. This involves calculation of the proba- 

 bility of deviations in one direction (beyond zero difference) from 

 the sample difference. If the sample difference of means is considered 

 as positive, then the negative "tail" of the theoretical frequency 

 curve of sample differences (this curve being centered at the observed 

 sample difference) must be compared with the rest of the curve. The 

 positive portion of the curve the ^ (1 -j- a)-* of the tables, then gives 

 the chances favoring the hypothesis that the sample means truly 

 represent the population means. The odds in favor of the hypothesis 

 are therefore given by the formula 



Values calculated from this formula are given in columns 4 and 5 of 

 table 40. 



"When other considerations than the sample evidence are to be taken 

 as determining the most probable value of the "true" mean, the case 

 is different. For example, if the probability that our sample per- 

 centages are mere sampling deviations from some theoretical Mendelian 

 value were in question, that theoretical value must be taken as the 

 population mean and only the magnitude of the deviations must be 

 considered. 



When a difference of means is considered from this latter stand- 

 point, it is assumed that the two samples come from one statistical 

 population, and hence that zero is the most probable value of the 

 population difference. If we choose to assume that the most probable 

 value of the population difference in our cases is zero, we must 

 calculate the odds against a deviation of the observed amount in 

 either direction from zero difference. The formula for these odds is 



'- 2XHl-a) 1-a- 



~* The whole area of tlie frequency curve is taken as unity, and a is the area 

 enclosed by any given deviation in both directions from the mean. 



