Histogram

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

An example histogram of the heights of 31 Black Cherry trees.

In statistics, a histogram is a graphical display of tabulated frequencies. A histogram is the graphical version of a table that shows what proportion of cases fall into each of several or many specified categories. The histogram differs from a bar chart in that it is the area of the bar that denotes the value, not the height, a crucial distinction when the categories are not of uniform width (Lancaster, 1974). The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent.

The word histogram is derived from Greek: histos 'anything set upright' (as the masts of a ship, the bar of a loom, or the vertical bars of a histogram); gramma 'drawing, record, writing'. The histogram is one of the seven basic tools of quality control, which also include the Pareto chart, check sheet, control chart, cause-and-effect diagram, flowchart, and scatter diagram. A generalization of the histogram is kernel smoothing techniques. This will construct a very smooth probability density function from the supplied data.

Contents

1 Examples

2 Activities and demonstrations

3 Mathematical definition

3.1 Cumulative histogram

3.2 Number of bins and width

4 Continuous data

5 See also

6 References

7 External links

[edit] Examples

As an example we consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, [1], Table 5). The census found that there were 124 million people who work outside of their homes.This rounding is a common phenomenon when collecting data from people.

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.

Data by absolute numbers

Interval | Width | Quantity | Quantity/width
0 | 5 | 4180 | 836
5 | 5 | 13687 | 2737
10 | 5 | 18618 | 3723
15 | 5 | 19634 | 3926
20 | 5 | 17981 | 3596
25 | 5 | 7190 | 1438
30 | 5 | 16369 | 3273
35 | 5 | 3212 | 642
40 | 5 | 4122 | 824
45 | 15 | 9200 | 613
60 | 30 | 6461 | 215
90 | 60 | 3435 | 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 shows absolute numbers.

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

Data by proportion

Interval | Width | Quantity (Q) | Q/total/width
0 | 5 | 4180 | 0.0067
5 | 5 | 13687 | 0.0220
10 | 5 | 18618 | 0.0300
15 | 5 | 19634 | 0.0316
20 | 5 | 17981 | 0.0289
25 | 5 | 7190 | 0.0115
30 | 5 | 16369 | 0.0263
35 | 5 | 3212 | 0.0051
40 | 5 | 4122 | 0.0066
45 | 15 | 9200 | 0.0049
60 | 30 | 6461 | 0.0017
90 | 60 | 3435 | 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%. The curve displayed is a simple density estimate. This version shows proportions, and is also known as a unit area histogram.

In other words a histogram represents a frequency distribution by means of rectangles whose widths represent class intervals and whose areas are proportional to the corresponding frequencies. They only place the bars together to make it easier to compare data.

[edit] Activities and demonstrations

The SOCR resource pages contain a number of hands-on interactive activities demonstrating the concept of a Histogram, histogram construction and manipulation using Java applets and charts.

[edit] Mathematical definition

In a more general mathematical sense, a histogram is a mapping mi 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 k be the total number of bins, the histogram mi meets the following conditions:

[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 Mi of a histogram mi is defined as:

[edit] Number of bins and width

There is no "best" number of bins, and different bin sizes can reveal different features of the data. Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. You should always experiment with bin widths before choosing one (or more) that illustrate the salient features in your data.

The number of bins k can be calculated directly, or from a suggested bin width h:

The braces indicate the ceiling function.

Sturges' formula

which implicitly bases the bin sizes on the range of the data, and can perform poorly if n < 30.

Scott's choice[1]

where h is the common bin width, and s is the sample standard deviation.

Freedman-Diaconis' choice[2]

which is based on the interquartile range

[edit] Continuous data

The idea of a histogram can be generalized to continuous data. Let (see Lebesgue space), then the cumulative histogram operator H can be defined by:

H(f)(y) = with only finitely many intervals of monotony this can be rewritten as

.

h(f)(y) is undefined if y is the value of a stationary point.

[edit] See also

Density estimation

Freedman-Diaconis rule

Image histogram

Kernel density estimation , another method of visualizing probability density functions that can be preferred to histograms.

[edit] References

^ Scott, David W. (1979). "On optimal and data-based histograms". Biometrika 66 (3): 605–610. doi:10.1093/biomet/66.3.605.

^ Freedman, David; Diaconis, P. (1981). "On the histogram as a density estimator: L2 theory". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57 (4): 453–476. doi:10.1007/BF01025868.

Webster's Third New International Dictionary, Merriam-Webster; Ind Una edition (June 2002).

Lancaster, H.O. An Introduction to Medical Statistics. John Wiley and Sons. 1974. ISBN 0 471 51250-8

[edit] External links

Journey To Work and Place Of Work (location of census document cited in example)

Understanding histograms in digital photography

Histograms: Construction, Analysis and Understanding with external links and an application to particle Physics.

A Method for Selecting the Bin Size of a Histogram

Retrieved from "http://en.wikipedia.org/wiki/Histogram"

Categories: Statistical charts and diagrams | Quality control tools | Non-parametric statistics

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This page was last modified on 14 May 2008, at 19:54.

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