Sec. 2.3 OTHER AVERAGES 21 



Under most circumstances it is easier to compute the geometric 

 mean from the relation 



(2.32) log(GM) =-Z(logX). 



N 



As an illustration consider the numbers 2, 5, 8, and 15. By definition 

 GM = the fourth root of the product (2) • (5) • (8) • (15) ; but, using 

 logarithms to the base 10, one has log GM = (log 2 + • • • + log 

 15) /4 = 0.7698. The antilog 0.7698 is approximately 5.9, which is 

 the geometric mean of the given set of numbers. The geometric 

 mean is useful in the calculation of certain index numbers, in studies 

 of biological growth, and, in general, whenever the statistical array 

 indicates that a geometrical series is involved. Obviously the geo- 

 metric mean is not used if any X — or if the product under the 

 radical is negative. 



The last average to be considered herein is called the harmonic 

 mean. It also is used only in specialized circumstances, but the pos- 

 session of some information about it will help to round out the reader's 

 knowledge regarding statistical averages. 



The harmonic mean is defined as the reciprocal of the arithmetic 

 mean of the reciprocals of a given group of numbers; or 



1 N 



(2.33) HM = 



\2(1/Xi)]/N 2(1/Z l -) 



For example, if the X's are 3, 8, 2, 5, and 2 the denominator is 

 2(1/Xi) = 1/3 + 1/8 + 1/2 + 1/5 + 1/2 = 1.6583, approximately; 

 hence HM = 5/1.6583 = 3.02. One use of the harmonic mean comes 

 when rates of some sort are involved. Consider this problem. A man 

 drives the first 50 miles of a trip at 50 mph, and the second 50 miles 

 at a rate of 60 mph. What is his average rate for the trip? By the 

 usual definition, the average rate is obtained by dividing the total 

 distance traveled by the total time taken to go that distance. The 

 distance traveled was 100 miles. The first 50 miles took one hour, 

 and the second 50 miles took five-sixths of an hour; hence the total 

 time was 11/6 hours. Therefore, the average rate of speed was 

 100/(11/6) = 600/11 = 54 and 6/11 mph. The harmonic mean of 

 50 and 60 also is 54 and 6/11 mph; that is, the required average rate 

 is just the harmonic mean of the two rates in this instance. It is 

 noted that the distance traveled was the same for each rate of speed. 

 Now suppose that a person drives for one hour at 50 mph and then 

 the second hour at 60 mph. What is the average rate of speed during 



