Sec. 2.3 OTHER AVERAGES 19 



basketball players differed from the mean weight of the group by 



(29 + 16 + 11 + 7 + 4 + 10 + 20 + 29)/8 



= 126/8 = 15.75 pounds. 



Then the average deviation for these weights is 15.75 pounds. 

 Symbolically the average deviation is defined by 



S| X - n I Si x I 



(2.21) AD = — -» or AD = 



N N 



where | x | = a deviation from fi taken as a positive number whether 

 the corresponding X was larger than [x or smaller than ju.. 



For the weights just used for illustration, a = 18.11 pounds. The 

 standard deviation is larger than the mean deviation, as is usual. 

 The standard deviation is much more widely used than the mean 

 deviation partly because it has many useful applications in sampling 

 studies, which after all is by far the more fruitful and interesting 

 field of statistical analysis. 



2.3 OTHER AVERAGES 



Another average which is simple to compute and of rather wide 

 application for descriptive purposes is the median, symbolized as md. 

 The median of a set of numerical measurements is intended to be a 

 number such that one-half the numbers are less than or equal to the 

 median, and the other half are greater than or equal to the median; 

 that is, the median is exactly in the middle of the set of numbers in 

 order of size, if such is possible. 



It is necessary — either actually or effectively — to list the numbers 

 in order of size before the median can be determined accurately. 

 Such an ordered group of numbers is called an ordered array. Thus 

 the numbers 1, 5, 2, 3, 0, 1, 8, and 10 do not form an ordered array, 

 whereas these same numbers listed as 0, 1, 1, 2, 3, 5, 8, and 10 do 

 constitute an ordered array. 



With the definition of an ordered array established, it is con- 

 venient to define the median of N numbers: X 1} X 2 , . . . , X N as the 

 [{N -f l)/2]th number in the array, starting with the lowest num- 

 ber in the array. It is noted that only if N is odd does such an ordinal 

 number exist; but it is sufficient herein to define an "ordinal" number 

 like 4.5 to be a number which is just midway in magnitude between 

 the fourth and fifth numbers in the array. For example, for the 



