20 SUMMARIZATION OF DATA Ch. 2 



array used above, N = 8, so that (N + l)/2 = 4.5. Hence the 

 median is md = 2.5, a number midway in size between the 2 (which 

 is the fourth number in the array) and the 3 (which is the fifth num- 

 ber in the array) . 



It can be seen, with a little study, that the median is an average 

 which will be nearer the region of concentration of the numerical 

 measurements in a population than the arithmetic mean if there are 

 a few "stray" numbers at one end of the scale of measurement. For 

 example, consider the following simulated annual salaries (in thou- 

 sands of dollars) of college instructors in one department: 3.1, 3.5, 

 3.5, 3.6, 3.6, 3.6, 3.8, 3.8, 3.8, 3.9, 3.9, 4.0, 4.0, 4.0, 4.4, 4.8, 5.0, 6.5, 

 8.4, 8.7, and 8.8. For these data N = 21, 2X = 96.1, /* = 4.7, and 

 md = 3.9. It is seen that eleven of that staff are receiving within 

 $300 of the median salary whereas only three are that close to the 

 arithmetic mean. The arithmetic mean exaggerates the typical salary 

 in a very real sense for all but the fortunate six at the top. In situa- 

 tions of this sort — which will be described later as skewed distribu- 

 tions when more data are involved — the median is a better average 

 than the arithmetic mean when its purpose is to describe the typical 

 measurement in the population. 



If a fairly large group of numbers is to be summarized and the 

 median is a desirable average to use, the midrange can be helpful in 

 reducing the necessary labor. For example, the MR for Table 2.01 

 is 103; hence we can hope that the median has about the same size. 

 On this assumption we can count the scores greater than or equal 

 to 100 and thereafter determine exactly the 645.5th number in the 

 array without excessive labor. It thus is found that md = 97. 



There are three other averages which will be considered, and 

 which will find occasional application to numerical measurements. 

 One is the mode (MO). The mode is defined to be that measure- 

 ment which occurs in a given set of numbers with the greatest fre- 

 quency, if such a number exists. For example, the mode of the set 

 5, 8, 9, 10, 10, 10, 11, 13, and 15 is MO = 10. If some number in a 

 group of data decisively occurs with the greatest frequency, the 

 mode may well be the average to use; but such is rather rarely the 

 case. 



The geometric mean of Xi, . . . , X N is defined as the iVth root of 

 the product of these numbers. Symbolically, 



(2.31) GM = VX 1 X 2 X 3 • • • X N 



