THE STATISTICAL STUDY OF VARIATION 47 



+ 3E the chances are 21 to 1 

 4E the chances are 142 to 1 

 5E the chances are 1310 to 1 

 QE the chances are 19,200 to 1. 



Since biometricians use the standard deviation as the measure of 

 variability, the relation between it and the quartile is utilized in deter- 

 mining all probable errors, even though there is some real error in such 

 a proceeding due to the distribution scarcely ever being exactly normal. 

 The probable error of the mean is found by multiplying the standard 

 deviation by 0.6745 and dividing by the square root of the number of 



0.6745<r 



variates, thus E m = /= ' Hence the reliability of the determi- 



V n 



nation of the mean increases not in proportion to the number of 

 variates but in proportion to the increase of their square roots. 



The probable errors of the standard deviation and the coefficient of 

 variability are as follows, but it is not necessary here to go into the proof 

 of the determinations. 



+ 0.6745cr 



Ea = ' 



+ 0.6745CT / ( 



I' + Hloo 



+ 0.6745C 



^ \^2n^ 



approximately if C is not greater than 10 per cent, because, if the group 

 of variates approximates a normal frequency distribution, the value of 

 C will be less than 10 per cent, and the value of the quantity within the 

 brackets will approximate unity and so can be neglected. 



The significance of probable error is most apparent when comparing 

 statistical results; for example, the standard deviations for average total 

 yield of plant in two or more varieties. Concerning the significance of 

 probable errors Rietz and Smith make the following statement: 



In the comparison of two statistical results, the difference between 

 the two results compared to its provable error is of great value. In 

 general, we may take the probable error in a difference to be the square root 

 of the sum of the squares of the probable errors of the two results. If the 

 difference does not exceed two or three times the probable error thus 

 obtained, the difference may reasonably be attributed to random sam- 

 pling. If the difference between the two results is as much as five to ten 

 times the probajble error, the probability of such differences in random 

 sampling is so small that we are justified in saying that the difference is 

 significant. In fact a difference of ten times its probable error is certainly 

 significant in so far as there is certainty in human affairs. 



