2-12. Absolute/Relative Frequency Distribution (Grouped Data)

An absolute frequency distribution shows the absolute frequencies with which the various values of a variable X are distributed among chosen classes. Here the absolute frequency of class j, fj , is the number of items falling into the jth class. For instance, an example of a typical absolute frequency distribution is:

Classes ofX fj fj N mj Class Boundaries 0–9 2 0.058 4.5 −0.5–9.5 10–19 7 0.205 14.5 9.5–19.5 20–29 10 0.294 24.5 19.5–29.5 30–39 8 0.235 34.5 29.5–39.5 40–49 6 0.176 44.5 39.5–49.5 50–59 1 0.029 54.5 49.5–59.5 34 1.000 The absolute frequency distribution consists of columns one and two, whereas a relative frequency distribution involves columns one and three, where the relative frequency of class j is fj N . For k classes, kj =1 fj = N and kj =1 fj N = 1. (If the data set is a sample of size n, then N is replaced by n in the preceding discussion.) In addition, the class mark of the jth class, mj , is the midpoint of the jth class and thus serves as a representative or typical value from the class; it is determined as the average of the class limits.
Next, class boundaries (sometimes called real class limits) are “impossible”
values and serve to avoid gaps between the classes. Finally, the class interval (denoted as

c) is the length of a class and is computed as the difference between the upper and lower boundaries of the class. (Note that if the classes are of equal length, then the class interval can be determined by taking the difference between two successive class marks.) Thus the common class interval for this distribution is, using the second class, 19.5–9.5 = 10 = c (which also equals m2 − m1 = 14.5 − 4.5 = 10). Given the n = 25 observations that follow, determine an absolute frequency distribution using the classes: 30–39, 40–49, . . . , 90–99. What are the relative frequencies, class marks, class boundaries, and the class interval?
33 87 39 40 44 44 97 48 53 50 67 37 51 66 94 55 55 68 71 76 83 87 70 89 90