Growth and variation in maize. 151 



distribution in which the variates are nearly equally distributed iu all 

 classes but in which there is a tendency towards a slight hump near 

 the middle class. 



Now, it can be shown that this is not at all the form of a distri- 

 bution wliich would be obtained if the observations were distributed 

 according to the laws of chance. It will be remembered that in the 

 above paragi'aph we are discussing the distribution of the mean quin- 

 tile position and not the distribution of the direct observations. 



The discussion can most readily be put into terms of probability 

 if we suppose ourselves to be dealing with throws of dice in place of 

 the corn measurements. Each corn plant was measured fourteen times. 

 Each measurement could fall in any one of five classes, i. e., in quin- 

 tile I, II, ni, etc. These conditions are strictly analogous to a throw 

 of fourteen five-faced dice. The chance of success, p, for any face is 

 1/5, and the mean number of dice sho"näng a given face at any throw 

 will tend to be np or 14/5. It is clear that if the dice are unbiased 

 this mean will be the same for each of the five faces. Thus the ob- 

 servations tend to be distributed equally among the several classes. 



Such a distribution is strictly analogous to the quintile distribu- 

 tions of the individual plants as given in tables 62 to 64. What has 

 been termed the mean quintile position would be analogous to the 

 mean number of spots on the fourteen dice at any throw. This mean 

 can be obtained from the frequency distribution given above in which 

 each of the five classes have equal frequency. In a distribution where 

 the frequency in each of n classes is the same, the mean in terms of 

 class units is equal to the mean of the first n natural number or the 



n + 1 



mean = — ^ — . 



When n = 5 the mean is equal to 3. 



It has been shown above that the standard deviation of the 

 first n natural numbers is given bj' 



„ ■ n2— 1 



12 

 If n = 5, then = V 2 = 1-4142. 



Thus the average number of spots per die showing at any throw 

 will tend to be 3 and the standard deviation of this mean will tend 

 to be V^. 



Now supposing we had a series of such throws each with a mean 

 approaching 3 but tending to deviate by a standard deviation of V 2. 



