STATISTICAL TREATMENT 237 



mean to be expected from sampling alone. This standard error is given 

 by the following equation : 



Standard error of mean 



Cm — 



V 



n 



where <r indicates the standard deviation of the individuals and n the 

 number of individuals. A similar treatment of the standard deviation 

 determines its standard error as follows: 



Standard error of st. dev. = cr^ = — -7=. 



These standard errors together with Table 3 may be used to indicate the 

 likelihood that the mean and standard deviation would vary by simple 

 sampling to any assigned degree. The numerical values of the standard 

 errors for the illustration under consideration are given at the bottom of 

 Table 5. 



A comparison of the average length of the irradiated and the non- 

 irradiated animals shows a small difference in favor of the irradiated 

 group, this group being on the average 0.899m longer than the non- 

 irradiated. To test the significance of this difference we follow the 

 procedure used for percentages and state that the standard error of this 

 difference is given by 



cdiff. = V<y' + <rl. -^ V(.047)2 + (.056)2 ^ o.073 



Turning to Table 3 with the value 



Diff. _ 0.899 

 cTdiff. ~ 0.073 



= 12.3 



we find the probability to be less than two in a billion that the observed 

 difference is due to simple sampling, and we are therefore willing to 

 proceed to a discussion of the reasons for this difference. Similar treat- 

 ment of the standard deviation shows that the irradiated group has a 

 standard deviation 0.194ju greater than the control, and the standard 

 error of this difference being 0.052, the difference is 3.7 times its standard 

 error. From Table 3 we find that the probability of such a difference 

 occurring by simple sampling is less than 2 in 10,000 and the difference 

 would therefore be considered significant. Thus the individuals in the 

 irradiated group are not only longer on the average than the nonirradiated 

 individuals, but they show a greater degree of absolute variation in 

 length. 



In addition to discussing the absolute variability of these forms it is 

 of interest to consider their relative variability and this is indicated by 



