EXPERIMENTAL DESIGN 221 



of the variance cr-/n corresponding to the standard error cr/\/n. The 

 variance of the distribution of means is 



(0.50)^ +(6.25)^ + (6.75)^ ^^,^, 



This is an estimate of crV4, so the estimate of cr^ is equal to 169.7. Since 

 we are deahng with three values, there are two degrees of freedom. Multi- 

 plying by the number of degrees of freedom gives us a "sum of squares" 

 which will be useful as a numerical check later. 



Table 16-3. Analysis of Variance of Data in Table 16-2 



Source of Variation Degrees of Freedom Sum of Squares Mean Square 



Individuals 9 417 46.3 



Treatments 2 339 169.7 



Total 11 756 68.7 



Now these figures are placed in an "analysis of variance" table (Table 

 16-3). Most of the numbers in this example are transferred directly from 

 Table 16-2 or the foregoing discussion. The next step is to compare the 

 variance estimates (called mean square in the table) by calculating a 

 ratio called F. 



mean square of sample means _ 169.7 



mean square of individuals 46.3 



The table of distribution of F values shows us that a value of F this large 

 occurs by chance more than 5 per cent of the time. In other words, the 

 variations in the treatments are not large enough to be statistically sig- 

 nificant, and it made no difference whether the plants were raised in 

 soil, vermiculite, or liquid culture. 



If the difference had been significant, further calculations, pinpoint- 

 ing the treatments which differed significantly would be possible. 



Factorial experiments 



It is often desirable to know the effects of several different factors on 

 the responses of biological materials. Each of these factors could be 

 tested separately by holding all the other factors constant. Temperature 

 and the concentration of glucose both might influence the respiration of 

 yeast cells, for example. Separate measurements of rate could be made at 



