22 SUMMARIZATION OF DATA Ch. 2 



this trip? The total distance traveled is 110 miles, and it took 2 

 hours; therefore the average rate is 55 mph. But that is just the 

 arithmetic mean of the two rates. It is seen that when time (hours) 

 was fixed in the problem, the appropriate average was the arith- 

 metic mean; when the distance (miles) was fixed and time was 

 variable according to the speed of travel, the appropriate average 

 was the harmonic mean. 



In general, the proper average to use in any particular situation 

 either will be determined at the outset by previous practices in the 

 particular sort of work, or it can be determined by a bit of prelim- 

 inary study of the matter. Hence no attempt will be made to lay 

 down rules. However, it should be apparent to the reader that when 

 a body of data is to be summarized statistically there may be several 

 possible choices of averages and also of measures of variation. We 

 should be fully conscious of this fact when we compute averages, or 

 when we interpret those averages computed by someone else. 



PROBLEMS 



1. The following numbers are salaries (in dollars) in a public school system 

 before World War II: 1300, 1500, 1300. 1350, 1600, 1250, 1400, 1350, 1800, 4500, 

 1450, 3000, 2200, 1250, 1300, 1550, 1700, 1600, 1350, 1400, 1450, 1750, 1500, 1600, 

 and 1400. Calculate the arithmetic mean and the median, and state which aver- 

 age you consider the more typical of these salaries. 



2. Suppose, in problem 1, that the following raises in salary were given: $2200 

 to $3000, $3000 to $3500, $4500 to $5000, $1800 to $2200, $1750 to $2200; and 

 all others are given a $100 raise. The salaries of problem 1 add to $41,850, and 

 the total of the raises is $4650, approximately 11 per cent of $41,850. Is it then 

 fair to state that those teachers received an 11 per cent increase in salary, on 

 the average? 



3. Compute the geometric mean of 76.3 and 85.1. 



4. Sometimes the median can be used as an average when numerical measure- 

 ments are not employed. For example, some radio direction finder networks 

 rate their bearings as to their quality, ranging from A, (best) through B, C, F, 

 and P. If the median does not turn out to be indeterminate (as by falling be- 

 tween two different letters) it may be useful in describing the average quality 

 of the readings. Obtain the median quality of the following quality ratings: 

 A, C, C, B, A, B, C, B, P, F, C, C, F, B, C, A, and B. Ans. C. 



5. What is the modal quality rating for problem 4? 



6. Compute the geometric mean of the salaries in problem 1 to the nearest 

 dollar. Ans. $1591. 



7. Suppose that peaches were bought in three different areas for $3 per bushel, 

 $2 per bushel, and $4 per bushel, respectively. Suppose also that $24 was spent 

 for peaches at each price level. What was the average price paid per bushel? 



8. Do as in problem 7 except to consider that 10 bushels of peaches were 

 bought at each price. Ans. $3 per bushel. 



