Example 1

Measurements of subscriber behaviour can be represented in a histogram. When subscriber A calls subscriber B the call may take on the following behaviour:

Thus for a particular exchange one can plot a histogram of failed call attempts which were repeated within 7 minutes when subscriber B is busy.

Example 2

This example is extracted from (Wikipedia 2006). All table data and figures are also taken from (Wikipedia 2006). Consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, Table 5). Actually, this document shows bar graphs, but they are not histograms since the bars are not adjacent. The census found that there were 124 million people who work outside of their homes. People were asked how long it takes them to get to work, and their responses were divided into categories: less than 5 minutes, more than 5 minutes and less than 10, more than 10 minutes and less than 15, and so on. The tables shows the numbers of people per category in thousands, so that 4,180 means 4,180,000.

The data in the following tables are displayed graphically by the diagrams below. An interesting feature of both diagrams is the spike in the 30 to 35 minutes category. It seems likely that this is an artifact: half an hour is a common unit of informal time measurement, so people whose travel times were perhaps a little less than or a little greater than 30 minutes might be inclined to answer "30 minutes".

Data by absolute numbers

Figure 1

Histogram of travel time, US 2000 census. Area under the curve (figure 1) equals the total number of cases. This diagram uses Q/width from the table 1.

Table 1: Used for figure 1.

This histogram shows the number of cases per unit interval so that the height of each bar is equal to the proportion of total people in the survey who fall into that category. The area under the curve represents the total number of cases (124 million). This type of histogram is ideal for an overview of absolute numbers.

Data by proportion

Figure 2

Histogram of travel time, US 2000 census. Area under the curve (figure 2) equals 1. This diagram uses Q/total/width from the table 2.

Table 2: Used for figure 2.

This histogram differs from the first only in the vertical scale. The height of each bar is the decimal percentage of the total that each category represents, and the total height of all the bars is equal to 1, the decimal equivalent of 100%. This version is ideal for comparing proportions. In a more general mathematical sense, a histogram is simply a mapping that counts the number of observations that fall into various disjoint categories (known as bins), whereas the graph of a histogram, which is often taught at high-school, is merely one way to represent a histogram (Wikipedia 2006). Thus, if we let N be the total number of observations and n be the total number of bins, the histogram hk meets the following conditions:

Figure 3

where k is an index over the bins (Wikipedia 2006).