APPENDIX TO CHAPTER XVI 407 
A simple measure of the extent of variability displayed by 
such a series of deviations from the mean is what is called the 
“standard deviation.” This is a sort of average of the devia- 
tions—not the ordinary arithmetical mean, but the mean 
found by taking the arithmetical mean of the squares of the 
deviations and extracting the square root. The standard 
deviation which represents the above twelve individual devia- 
tions is thus, — 
+ (= 1.924 (—.9) + (1)? 
CDE CDR (LDR (124 (2h (= 954 (Dy bie 1 (1) 
12 
which is .95. 
This “standard deviation” is used instead of other averages 
for several reasons. The arithmetical mean of the deviations 
about the mean is quite unavailable, because, unless it be 
reckoned by disregarding all minus signs (in which case the 
result is an illogical makeshift), it is zero; the standard 
deviation is very readily calculated, not by performing the 
operations indicated above, but by recourse to a theorem 
that the mean of the squares of the deviations about the 
mean is equal to the mean of the squares of the deviations 
about any other magnitude less the square of the difference 
between the mean and this other magnitude. The proof 
of this theorem is simple, and may be found in the books 
on probability. Applying it to the illustrated case, we first 
take the deviations, not about the mean, but about some other 
magnitude, say 5%. These deviations are 0,0,1,0,0, —1,0, 
2,0, —2, —1,0. The squares of these are 0,0,1,0,0,1,0,4, 
0, 4, 1, 0, of which the arithmetical mean is 4H, or .902. This 
is the mean of the squares of the deviations about the mag- 
nitude 5. From this we are to deduct the square of the 
difference between the mean 4.9 and the other magnitude 5, 
about which the deviations were measured. The difference is 
1, its square is .01. Deducting this from .902 we have .892 as 
the mean of the squares of the deviations about the mean. The 
Square root of this is .95, which is therefore the standard 
deviation sought. Calculated by this method the standard 
deviation may usually be obtained in less than one tenth the 
time required by the direct method. 
    
    
   
   
   
   
   
  
     
  
  
  
  
  
  
  
   
    
  
  
  
  
  
   
  
   
  
  
TR RR A sk 
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