Sec. 2.1 ARITHMETIC MEAN AND STANDARD DEVIATION 13 



An average called the arithmetic mean has been found to possess 

 all the properties (2.11) to (2.15) to a rather high degree for a broad 

 class of statistical populations. In addition, it is extremely useful in 

 the analysis of sampling data, as will be shown later. Hence the 

 arithmetic mean is a highly recommended average. 



The arithmetic mean, jx (Greek letter mu), of N measurements: 

 Xi, . . . , X N is calculated by dividing the sum of the N measurements 

 by N. Symbolically, 



(2.11) n = 



x 1 + x 2 + x 3 +---+x 



N 



N 



N 

 or for brevity, y. = 



N N 



To illustrate, suppose that X x = 2, X 2 = 5, X 3 = 1, X 4 = 3, and X 5 

 = 4; then the arithmetic mean is 



M = (2 + 5+1+3 + 4)/5 = 3.* 



Problem 2.11. Suppose that eight players are on the traveling squad of a 

 basketball team, and their weights are 152, 170, 165, 185, 201, 174, 191, and 210 

 pounds. What is the arithmetic mean of these weights? 



The first question which may occur to the student is: What are the 

 Xi in this instance? It is a well-known assumption in arithmetic 

 and algebra that the same sum is obtained for a given set of numbers 

 no matter what the order of addition; that is, 3 + 6 + 15 = 24 = 

 6 + 15 + 3 = 15 + 3 + 6, or any other possible order of addition. 

 Likewise, in the present problem, it makes no difference which weight 

 is symbolized as X x , which as X 2 , etc. It is convenient just to let 

 Xi = the first weight listed, X 2 = the second weight on the list, and 

 so on. If that is done in this problem, X\ = 152, X 2 — 170, X 3 = 165, 



* Although the discussion in this chapter is chiefly devoted to methods and 

 ideas appropriate to populations of data — which usually contain a large number 

 of measurements — small groups of numbers will be used in examples and prob- 

 lems for the purpose of facilitating and shortening discussions. Obviously, most 

 of these problems and examples resemble samples far more than populations. 

 However, the methods introduced will apply to populations and will not neces- 

 sarily be correct or efficient for sampling studies, as will be noted later in the 

 book. 



