28 SUMMARIZATION OF DATA Ch. 2 



9. Within what extremes did the lowest one-fourth of the counts listed in 

 problem 6 lie? The middle one-fourth? The highest one-fourth? 



10. Take any newspaper with at least one hundred bond or stock quotations 

 and make frequency and relative cumulative frequency distributions of those 

 prices. 



2.5 CALCULATION OF THE ARITHMETIC MEAN AND 



THE STANDARD DEVIATION FROM FREQUENCY 



DISTRIBUTION TABLES 



If the frequency distribution table has class intervals of equal 

 lengths, approximate values can be computed for ^ and o- with a 

 considerable saving in labor as compared to their computation from 

 the individual measurements. The method of computation involves 

 the sole assumption that the numbers grouped into each class actu- 

 ally were at the midpoint of their class. Although that assumption 

 is not strictly correct, the individual discrepancies usually balance 

 out so well that the net error is unimportant in practice. If it should 

 be decided that some additional accuracy is needed, Sheppard's cor- 

 rections for grouping can be employed. (See, for example, Kenney, 

 Mathematics of Statistics, Part One, D. Van Nostrand.) 



Table 2.51 presents methods for computing ^ and a which follow 

 directly from the definitions of these quantities if all the data in a 

 class are considered to be at the midpoint of the class. For example, 

 the data in Table 2.51 would be considered to be 22.5, 22.5, 17.5, 17.5, 

 17.5, 17.5, 17.5, 17.5, 12.5, 12.5, 12.5, 12.5, 12.5, 12.5, 12.5, 12.5, 12.5, 

 12.5, 7.5, 7.5, 7.5, 7.5, 7.5, 2.5, and 2.5 each midpoint appearing pre- 

 cisely that number of times indicated by the class frequency, /. The 

 student should check the fact that the sum of these products is 317.5, 

 which is shown in Table 2.51 as the total of the column headed "f-z." 

 The symbol z is employed to denote the midpoint of the class interval. 

 For convenience and for uniformity of procedure, the midpoint of a 

 class of data measured on a continuous scale is defined to be the 

 lower limit of the class (as recorded in the table) plus one-half the 

 length of the class interval. Also, the length of the class interval, for 

 such data, is defined as the numerical difference between any two 

 successive left- or right-hand end points of classes. Thus the mid- 

 point of the class "20-24.999 . . ." is 20 + (1/2) (5) = the z for this 

 class interval. 



The reader should note that there will be circumstances in practice 

 which will justify a different determination of the midpoint, z. For 



