Histogram

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For the histogram used in digital image processing, see Color histogram.

In statistics, a histogram is a graphical display of tabulated frequencies. 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.

The histogram is one of the seven basic tools of quality control, which include the histogram, Pareto chart, check sheet, control chart, cause-and-effect diagram, flowchart, and scatter diagram. See also Quality Management Glossary.

1 Examples
- 1.1 Data by absolute numbers
- 1.2 Data by proportion
2 Mathematical Definition
- 2.1 Cumulative Histogram
- 2.2 Rules of Thumb
3 See also
4 External links

[edit] Examples

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, [1], 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".

[edit] Data by absolute numbers

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

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.

[edit] Data by proportion

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

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 area of all the bars is equal to 1, the decimal equivalent of 100%. This version is ideal for comparing proportions.

[edit] 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 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 $h k$ meets the following conditions:

$N = \sum_{k=1}^n{h_k}$

where k is an index over the bins.

[edit] 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 $H k$ of a histogram $h k$ is defined as:

$H_k = \sum_{k\prime=1}^k{h_{k\prime}}$

[edit] Rules of Thumb

Most people use 7 to 10 classes for histograms. However, from time to time the following rules of thumb have been used to chunk the data. Where n is the number of observations in the sample.

$N_{class} = \sqrt n$

$N_{class} = A \sqrt n$

$N c l a s s = 10log n$

...with varying degrees of success. The final technique does not perform well with n < 30.

The bin width $Δ$ (or the number of bins) of a histogram can be selected by using the formula $(2 k - v) / Δ 2$ , where $k$ and $v$ are the mean and variance of the number of data points in the bins. The optimal bin width is the one that minimizes the formula. [2]