Growth and variation in maize. 123 



numbers iu each quiutile. That is, siuce p, tlie chance of success, is Vs, 

 and n is 143, the mean number falling in each iiuintile would be 

 np = 28 '6 observations, or 20 percent. 



In order to make the following discussion clear we may consider 

 its relation to throws of dice. If onl}' chance were acting on the corn 

 plants the results would be exactly comparable with those obtained by 

 making 11 throws of thirteen unbiased 5 -faced dice. The expectation 

 here for the mean number of successes of each face would be 28 "6 or 

 20 percent as given above. 



Further the standard deviation of this result would l)e 



Ö = Vnpq 



= V 143 X Vs X Vs 

 = 4-78 or 3-34 percent. 

 All or practically all the observations ought to fall within a range of 

 six times the standard deviation. Thus, if no influence other than chance 

 were acting, no observation above 30 percent and none below 10 per- 

 cent would be expected with very few deviating so far as these figures 

 from the mean. 



It is, of course, possible to observe the actual deviation of these 

 corn plants from this theoretical mean value. The theoretical mean 

 furnishes a base from which can be measured the tendency of plants 

 to remain relatively small, or relatively large, as the case may be, 

 throughout their life histories. 



Before discussing these deviations however it is necessary to note 

 that in the majority of these quintile distrilnitious p is not Vs, but 

 differs slightly from this value. Thus in Talile 7 there are in all 

 702 observations. Of these, 145 fall in ([uintile I. Thus the chance 

 of success, p, for observations falling in quintile I is '^'/tos, which 

 reduced to percentage gives 20*66 percent, instead of 20"00 per- 

 cent. Again in quintile V there are only 130 observations which gives 

 a theoretical mean of 18 '52 percent. As already stated, these deviations 

 are due to the fact that all the quintiles are not of exactly equal areas. 

 This method of calculating the probability makes allowance for these 

 deviations. In Tables 7 to 9 the figures in the percentage columns in 

 the last row of each table give these theoretical mean percentages on 

 the basis of pure chance. 



The standard deviation 



Ö — Vnpq 



