152 Unicerfiitji of California PiihUcatinits in Afirirnliural S"icnccN \ Vol. 4 



Values irum this t'ormula air given in columns 2 and 'A of table 40; 

 their magnitude in three oases, liowever, and the uniforiii agreement 

 of the direction of difference with the expectation from biological 

 evidence which has been discussed, weigh heavily in each test against 

 the assumption of random sampling from a single statistical population. 



It does not appear necessary, however, thus to weigh the evidence 

 in detail before deciding which formula is suited to the ease. There 

 is no evident theoretical value t'l-om which these percentages are 

 reasonably likely to be sampling deviations. This being the case, and 

 granting such general possibilities as that of differential viability, it 

 seems most reasonable to use the former (OJ formula. That is, we 

 should give full weight to the implications of a sample deviation 

 unless there is some definite reason for assuming that some other value 

 better represents the mean of the theoretical statistical population. 



It must be remembered that the actual probabilities of sampling 

 deviations do not necessar-ily correspond closely with the probabilities 

 of random sampling. With the material in table 40, however, aside 

 from the germination comparison in the case of the slender type, 

 table 39 suggests a fair agreement with the conditions of random 

 sampling. The actual standard deviations of the subsamples do not 

 in general diff'er widely from the corresponding theoretical values, and 

 the differences are negative about as often as positive. 



The hypothesis of selective elimination with poor germination is 

 strongly sustained (table 40), although only one difference (with the 

 crenate type) has much statistical significance when considered alone. 

 If we may multiply together the members of the four ratios in column 

 3 of the table, the combined odds (using the /„ values) are 130:1 

 against (iceurrence of these four deviations as accidents of simple 

 sampling, when magnitude of deviation alone is considered. If 

 direction of deviation alone is considered the random chance of these 

 four deviations all in the same direction is obviously (^)^, or the odds 

 favoring the elimination hypothesis are 15:1. Combination of these 

 two chances indicates a high probability for the hypothesis. When 

 the two-population formula is used in eal(;ulating the standard devia- 

 tion of the difference (columns 4 and 5) the value of P is consider- 

 ably reduced in some cases, and the combined odds obtained from 

 l\- F..- F..- 1\ are very high. Evidently the best single expression 

 of the simple-sampling odds, though po.ssibly somewhat too high, is the 

 value given last in column 5, or 123,093:1. 



With the seed-size test of crenate the odds are 499 : 1 with the 

 theoretical standard deviation of the difference, or 1666:1 with the 



