statistical Tests 163 



that complete linkage is the cause if, as in the old gold-bullata 

 cross, two widely different parts of the plant or animal are 

 affected in the different phenotypes. Even this distinction is not 

 always valid, however, for occasionally members of a series of 

 multiple alleles affect principally different regions of the or- 

 ganisms. 



Statistical Tests 



In Chapter 9 it was shown that x" niay be used to determine 

 whether an observed ratio could be considered an example of a 

 9:3:3:1 or other ratio involving more than two terms. If 

 an observed ratio has a value of x" low enough to be considered 

 a 9 : 3 : 3 : 1 ratio, we assume that we have independent assort- 

 ment and that the genes are on separate chromosomes; but if 

 X" is too large, we must find another explanation. 



In their early work on the sweet pea, Bateson and Punnett 

 found that purple flowers (B) were dominant over red (6) and 

 that long pollen grains (L) were dominant to round (I). A 

 homozygous purple-flowered, round pollen plant when crossed 

 with one that was homozygous for red flowers and long pollen 

 gave the results tabulated below (data from Punnett, 1913) ; x^ 

 is calculated on the basis of independent assortment: 



On the basis of an expected dihybrid 9:3:3:1 ratio this 

 particular observed ratio showed a x^ value of 32.36. As there 

 are four terms in this ratio, three degrees of freedom should 

 be used in calculating probability from Fisher's table (Table 3). 

 From the table we learn that a x^ oi 32.36 with three degrees 

 of freedom would occur in less than one per cent of similar 

 experiments as the result of chance alone. We must therefore 

 conclude that this ratio is not a true example of a 9 : 3 : 3 : 1 

 ratio and that we are not dealing with a case of independent 

 assortment. If these statistical methods tell us we do not have 



