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 



TABLE IV. To COMPUTE THE MEAN TOTAL YIELD 

 OF PLANT IN GRAMS. Let G = assumed mean = 3.5 



of type used and its value 

 should always be indicated 

 on a graph. For a precise 

 description of the variation 

 within a group it is neces- 

 sary to have something 

 more than a measure of the 

 type. Knowing the arith- 

 metical average is not 

 sufficient to permit com- 

 parison of the variation in 

 different populations. 

 There is needed some mea- 

 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 2 ), multiply each d 2 by the frequency (/), 

 summate the products, divide by n and extract the square root. This 

 is expressed by the formula 



<r = 



