THE STATISTICAL STUDY OF VARIATION 41 
variations from the true mean of all the items in the table equals zero. Unlike 
the mode it is affected by every item in the group so that its location 
can never be due to a single class; moreover it gives weight to extreme 
deviations. The measure 
of type used and its value Taste 1V.—To Comrutn tun Mnan Toran Yirip 
enol always beindicared) °8 Puantin Grams. Let G = assumed mean = 3.5 
on a graph. For a precise Vv f | ro | S(V-G) 
description of the variation a ener 
within a group it is neces- 0.5 3 | —3 - 9 
sary to have something 1.5 50 -2 — 100 
more than a measure of the ee ns Fe eae = 
type. Knowing the arith- 45 80. 1 80 
metical average is not 5.5 42 9 84 
sufficient to permit com- 6.5 7 3 21 
parison of the variation in 7.5 2 4 8 
different populations. aS : ? wae: 
There is needed some mea- Pea OO | 17 
sure of variability. 
The Standard Deviation, 
Calculation and Significance.—Examination of the original records 
of weighings of the total yield of the 400 oat plants would reveal 
a certain amount of variation in the yield of each plant from the 
mean yield, 3.458 g. The plants were grouped into classes in com- 
puting the mean yield and they can be treated similarly in calculating 
the average amount of variation from the mean yield for the whole 
sample. It may be noted that the simplest measure of the absolute 
variation within the sample is the average deviation, which is simply cal- 
culated by summating the products of the deviation of each class from 
the true mean multiplied by its frequency and dividing this sum by n. 
The standard deviation is universally preferred as an absolute measure of 
variability. The standard deviation differs from the average deviation 
in one important feature, viz., that in calculating the standard deviation 
each individual variation from the mean is squared. This gives addi- 
tional weight to the extreme variations which is especially desirable in 
biometrical work. 
In calculating the standard deviation (Table V) the regular procedure 
is as follows: Write the minus and plus deviation (d) of each class from the 
mean, square each deviation (d?), multiply each d? by the frequency (f), 
summate the products, divide by n and extract the square root. This 
is expressed by the formula 
= qeu ue), 
n 
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