STATISTICAL TREATMENTS 207 



the value of t, look up the probability in the table, and then draw con- 

 clusions in the same manner as in the normal deviate test. 



The Chi-square (x") Test: In some kinds of experiments the data 

 occur as discrete numbers rather than as continuously variable quanti- 

 ties. A genetics experiment might yield flowers that are either red or 

 white, and the observation of the results consists of counting the two 

 types. According to hypothesis, the two types should occur with a certain 

 probability. For example, we might expect three reds for each white; the 

 probability for red is %, and that for white is Vi. The X" test is an ap- 

 proximation of a much more involved test for deciding how well the 

 observed counts fit the hypothesis. 



A X^ value is calculated for each class of events. In the experiment we 

 observe a certain number of white flowers (o), but from the hypothesis 

 we can calculate the most probable number (c). The X" value is the 

 square of the deviation from the expected value divided by the expected 

 value : 



^ C 



A similar calculation is made for each of the classes. The final X" value 

 is the sum of these individual numbers 



,^ Co-cy I Co - cy 



C(white) C(red) 



In this test, only one degree of freedom exists because if a flower is 

 not white, it has to be red. In another experiment, flowers might be red, 

 white, or pink; here there are two degrees of freedom. 



A table of X" values is used in tests for significance. If the value of 

 X^ is larger than could be expected (P = 0.05 or O.OI) for this number 

 of degrees of freedom, then the observed data do not fit this hypothesis. 

 In the example the expectation was V4 white and 44 red. If the X" value 

 suggests that this hypothesis is incorrect, we could test some other hypoth- 

 esis, such as Vi white and Vi red. 



Analysis of variance 



In the ideal controlled experiment all factors are constant except the 

 one being tested. The significance of this varying factor is easily tested 

 by one of the foregoing statistics. In practice, however, the effect ob- 



