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 convex to concave;it therefore measures the 
slope of the curve and is a good measure of its variability. Measuring 
from M too 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 
+ 20 lie 95.5 per cent. of all individuals and within + 3c he 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 + 3c. 
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 + 4@Q includes 99.3 per cent. of the individuals 
M + 5@ 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 = 0.6745c. 
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 error is not the “‘most probable error.” 
The most probable error is 0 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,! 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 
+ 2K the chances are 4.6 to 1 
1The Germans use o 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 + 2o lie 95.5 per cent. of the variates 
within M + 3o lie 99.7 per cent, of the variates. 
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