Statistical Inference: A Short Course

2.4 Appendix 2.A Histograms with Classes of Different Lengths

We previously defined a histogram as a set of vertical bars whose heights correspond to absolute class frequencies and whose bases are the class intervals. This definition implicitly assumes that all of the classes are of equal length. However, for some data sets, this might not be the case.

A more general definition of a histogram is as follows: a set of vertical bars whose areas are directly proportional to the absolute class frequencies presented, where

(2.A.1)

and

(2.A.2)

Hence, frequency density amounts to absolute class frequency on a per standard width basis—it is the absolute class frequency per interval length. So for classes of equal length,

and for classes of different lengths,

Given Equation (2.A.2), we may now express Equation (2.A.1) as

(2.A.3)

where the constant of proportionality mentioned in the preceding definition of a histogram is

Example 2.A.1

Table 2.A.1 involves an absolute frequency distribution and ...