In statistics, a histogram is a graphical display of tabulated frequencies. That is, a histogram is the graphical version of a table which shows what proportion of cases fall into each of several or many specified categories. The categories are usually specified as nonoverlapping intervals of some variable. The categories (bars) must be adjacent.

Part of Seven tools of quality (Quality improvement tools that include the histogram, Pareto chart, check sheet, control chart, cause-and-effect diagram, flowchart, and scatter diagram.) see [[1]]

## Histogram Examples

There are many different ways to display the same table, and two kinds of histograms are shown below. As an example we consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, [2], 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

 Interval Width Quantity Quantity/width 0 5 4,180 836 5 5 13,687 2,737 10 5 18,618 3,723 15 5 19,634 3,926 20 5 17,981 3,596 25 5 7,190 1,438 30 5 16,369 3,273 35 5 3,212 642 40 5 4,122 824 45 15 9,200 613 60 30 6,461 215 90 60 3,435 57

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

 Interval Width Quantity (Q) Q/total/width 0 5 4,180 0.0067 5 5 13,687 0.0220 10 5 18,618 0.0300 15 5 19,634 0.0316 20 5 17,981 0.0289 25 5 7,190 0.0115 30 5 16,369 0.0263 35 5 3,212 0.0051 40 5 4,122 0.0066 45 15 9,200 0.0049 60 30 6,461 0.0017 90 60 3,435 0.0004

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.

## Mathematical Definition

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. Thus, if we let N be the total number of observations and n be the total number of bins, the histogram meets the following conditions:

where k is an index over the bins.

### Cumulative Histogram

A cumulative histogram is a mapping that counts the cumulative number of observations in all of the bins up to the specified bin. That is, the cumulative histogram of a histogram is defined as: