46 GENETICS IN RELATION TO AGRICULTURE 



its shape represents the general distribution of these individuals. The 

 mean represents the average of the distribution. The standard deviation 

 (plus and minus) represents the ordinates of those points on the curve 

 where the slope changes from con vex to concave; it therefore measures the 

 slope of the curve and is a goocTmeasure of its variability. Measuring 

 from M to o- on each side of the curve, we find that the space enclosed 

 includes 68.3 per cent, of the total number of individuals; within the limits 

 2<r lie 95.5 per cent, of all individuals and within + 3<r lie 99.7 per 

 cent. Thus we see that although theoretically the curve never meets 

 the ground line but extends out to infinity, practically all individuals are 

 found within the limits 30-. 



Similarly we find that the quartile measures the number of individuals 

 within the limits of the curve that it marks off as follows : 



M Q includes 50.0 per cent, of the individuals 

 M 2Q includes 82.3 per cent, of the individuals 

 M + 3Q includes 95.7 per cent, of the individuals 

 M 4Q includes 99.3 per cent, of the individuals 

 M 5Q includes 99.9 per cent, of the individuals. 



In a normal curve, therefore, the standard deviation and the quartile 

 have a constant relationship such that Q = 0. 67450-. 



From these relationships an idea of the meaning of the term "prob- 

 able error" which is always calculated for any series of observations may 

 be obtained. The probable error tells us what confidence we may place 

 in our work, if the errors are due to chance only and not to avoidable 

 mistakes of method. The probable erroris not the "most probable error." 

 The most probable error is and hence is identical with the mean. 

 Probable error is an arbitrary term used to denote the amount that must 

 be added to or subtracted from the observed value to obtain two limiting 

 figures of which it may be said that there is an even chance that the true 

 value lies within or without these limits. 



The probable error, E, of a single variate is the quartile, 1 since the 

 chances are even that any variate lies within or without the value M + Q; 

 and since 82.3 per cent, of the variates lie within the value M + 2Q, the 

 chances are 4.6 to 1 that the true value of any series of a calculated con- 

 stant is within these limits. Thus the chances that the true value lies 

 within any multiple of E are 



+ E the chances are even 

 2E the chances are 4.6 to 1 



1 The Germans use a as the measure of error. It is known as the error of mean 

 square and is proportionately larger than the probable error as is shown by the fact 

 that 



within M a lie 68.3 per cent, of the variates 

 within M 2<r lie 95.5 per cent, of the variates 

 within M 3a lie 99.7 per cent, of the variates. 



