292 MISCELLANEOUS STUDIES 



In the two cases (smooth-leaved and crenate-leaved types) where 

 the percentages of mutant types differ greatly with good and poor 

 germination, separation according to germination gives a mean value 

 of the standard deviation decidedly lower than the value for all lots 

 taken together. In the case of the large-leaved type there is little 

 change, while the considerable reduction with the slender type is 

 probably due to unequal separation of lots from parents genetically 

 different. 



Table 40 shows the simple-sampling probability of the most striking 

 differences of heredity percentages, aside from the characteristic 

 differences between different types. "Student's" (1917) table of 

 probabilities of mean deviations with small samples is used, with 

 interpolation by second differences. Where the standard deviation 

 of the difference is required it is found from the theoretical values 

 given in table 39 by the formula (Yule, 1911, pp. 264-265) 



-/ I _1^ ^0 ^0 I 

 <T difference V "l ~T~ 2 " 5"T 



Po Qo , Po Qo 



O I O 



when one statistical population is assumed (table 40, columns 2 and 3). 

 When two populations are assumed (table 40, columns 4 and 5) the cor- 

 responding formula using p l q 1 and p 2 q 2 is employed. In the one case 

 where this is possible (the seed-size test), it is also calculated from the 

 actual differences of the pairs of percentages in the separate tests, each 

 difference being weighted with the total number of progeny from the 

 parent concerned. Where two values of / (the n of "Student's" 

 table) are involved, the smaller is taken, giving understatements of the 

 probabilities involved; in the two cases where the difference is more 

 than 2, the values are recalculated, with / as the nearest smaller 

 integer to the geometric mean of the two actual numbers (that is 

 with /= V/i/2)- 1 the case where the probabilities of four devia- 

 tions all in the same direction are combined, the four chances of 

 occurrence are multiplied together; that is, if the (l-f- a ) of 

 "Student's" table is P, and l P is F, then F l . 2 . 3 . t = F^F^-F^F^ 

 "Student" (1908, p. 1) says, "The usual method of determining 

 the probability that the mean of the population lies within a given 

 distance of the mean of the sample, is to assume a normal distribution 

 about the mean of the sample . . . ." When this is done with a differ- 

 ence of means, it is at once evident that only half of the chances of 

 deviations as great as the distance of the given difference from zero 

 difference lie below zero difference; the other half of the chances of 



f!50] 



