126 Pearl and Surface. 



From the method of obtaining the theoretical mean percentages 

 (p. 122) as given in Tables 7 to 9, it is evident that the area of the 

 frequenc}' polygon obtained by plotting these mean values must be the 

 same as the area of the polygon obtained by plotting the observed 

 percentage values given in any row of the table. Thus, if, in the upper 

 left hand diagram of Fig. 6, we draw perpendiculars from the ends of 

 the line representing the values of the theoretical means, the area of 

 the enclosed polygon will be the same as that of the polygon formed 

 in a like manner from the line joining the observed percentage values. 

 In other words, the sum of the positive deviations from the theoretical means 

 is equal to the sum of the negative denations from those means. It 

 therefore follows that the sum of the squared deviations is a minimum 

 in every case. The root -mean -square deviation will then form an 

 excellent measure of the actual deviation of the observed values from 

 the values to be expected on the theory of probability. 



The root -mean -square deviation expresses in one constant the 

 amount by which all of the plants starting in a given quintile deviate 

 from what the mean value would have been under the conditions of 

 simple sampling. In other words, it measures the effect of whatever 

 influences other than chance are acting upon the given group of plants, 

 in respect to the determination of their place in the variation curve of 

 the population at different growth stages. 



The values of the root -mean -square deviations for the plants 

 starting in each quintile are given in the next to the last column of 

 tables 7 to 9. These constants for tlie three series are shown grai)bi- 

 callj' in Fig. 8. 



In order to estimate the significance of these root -mean -square 

 deviations it is necessary to know how widely such constants may 

 fluctuate under the conditions of simple sampling. If, for example, we 

 had thrown five-faced dice instead of measuring corn plants the per- 

 centage of successes would have been close to the theoretical mean of 

 20 percent. Howewer, unless the number of throws had been very 

 large the observed values would not have exactly coincided with the 

 theoretical values. The amount of the deviation wliich can be expected 

 in any instance is measured by the standard deviation of the theoretical 

 mean. Now it will have been noticed that the root- mean -square 

 de\iation discussed above is the standard deviation of the observations 

 taken about the theoretical mean, because the sum of the positive and 

 negative deviations from these means is zero. Therefore the standard 



