In statistics, a histogram is a graphical display of tabulated frequencies, shown as bars. It shows what proportion of cases fall into each of several categories: it is a form of data binning. The categories are usually specified as non-overlapping intervals of some variable. The categories (bars) must be adjacent. The intervals (or bands, or bins) are generally of the same size.
Histograms are used to plot density of data, and often for density estimation: estimating the probability density function of the underlying variable. The total area of a histogram used for probability density is always normalized to 1. If the length of the intervals on the x -axis are all 1, then a histogram is identical to a relative frequency plot.
An alternative to the histogram is kernel density estimation, which uses a kernel to smooth samples. This will construct a smooth probability density function, which will in general more accurately reflect the underlying variable.
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.
Etymology
The word histogram derived from the Greek histos 'anything set upright' (as the masts of a ship, the bar of a loom, or the vertical bars of a histogram); and gramma 'drawing, record, writing'. The term was introduced by Karl Pearson in 1895.
Examples
As an example we consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, , Table 2). The census found that there were 124 million people who work outside of their homes. An interesting feature of this graph is that the number recorded for "at least 30 but less than 35 minutes" is higher than for the bands on either side. This is likely to have arisen from people rounding their reported journey time. This rounding is a common phenomenon when collecting data from people.
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.
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.
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.
Mathematical definition
In a more general mathematical sense, a histogram is a mapping m i 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 m i meets the following conditions:
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 M i of a histogram m i is defined as:
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.
which implicitly bases the bin sizes on the range of the data, and can perform poorly if n < 30.
where σ is the sample standard deviation.
which is based on the interquartile range.
Continuous data
The idea of a histogram can be generalized to continuous data. Let ƒ ∈ L 1 ( R ) (see Lebesgue space), then the cumulative histogram operator H can be defined by:
h ( ƒ )( y ) is undefined if y is the value of a stationary point.
See also
- Data binning
- Freedman–Diaconis rule
- Image histogram
Density estimation
- Density estimation
- Kernel density estimation, a smoother but more complex method of density estimation
Notes
- ^ Howitt, D. and Cramer, D. (2008) "Statistics in Psychology". Prentice Hall
- ^ M. Eileen Magnello (December 1856). "Karl Pearson and the Origins of Modern Statistics: An Elastician becomes a Statistician". The New Zealand Journal for the History and Philosophy of Science and Technology 1 . ISSN 1177–1380 . http://www.rutherfordjournal.org/article010107.html .
- ^ Sturges, H. A. (1926). "The choice of a class interval". J. American Statistical Association : 65–66.
- ^ 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: L 2 theory". Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 57 (4): 453–476. doi: 10.1007/BF01025868 .
References
- 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
External links
- Journey To Work and Place Of Work (location of census document cited in example)
- Understanding histograms in digital photography
- Histograms:
Histogram - Wikipedia, the free encyclopedia
In statistics, a histogram is a graphical display of tabulated frequencies, shown as bars. It shows what proportion of cases fall into each of several categories: it is a form of ...
Generic Image Library : Histogram Example
Histogram Example Actual commercial code that computes the luminosity histogram (variable names have been changed and unrelated parts removed):
Histograms: Example #2
Histograms: Example #2. Objective: Visually explore the heights of a sample of students using a histogram. Characterize the shape of the distribution and assess how the ...
File:Histogram example.svg - Wikipedia, the free encyclopedia
Histogram_example.svg (SVG file, nominally 361 × 289 pixels, file size: 6 KB)
Histogram Examples:A picture of your data
This page provides some histogram examples. We discuss normal distribution and how it applies to quality assurance. Histograms are a key process improvement tool.
File:Histogram example.svg - Wikimedia Commons
[edit] R source #R version 2.3, www.r-project.org library("RSvgDevice") devSVG("histogram.svg", width = 5, height = 4) set.seed(50) par(mar=c(3,2,1,1),yaxs="i",mgp=c(1.8,.9,0 ...
Histogram Definition, Histogram Examples | TutorVista
Hundreds of tutors are online and ready to help you right now!
File:Histogram example.svg - Wikibooks, collection of open-content ...
[edit] R source #R version 2.3, www.r-project.org library("RSvgDevice") devSVG("histogram.svg", width = 5, height = 4) set.seed(50) par(mar=c(3,2,1,1),yaxs="i",mgp=c(1.8,.9,0 ...
Histogram Example
Histogram Examples. An intensity histogram is a graph, plotting the number of pixels (or fractional area) with a specific gray level vs. the gray level value.
Histogram Example
Histogram Example Histogram Matching with DOQs Version 1.0 Mark Lucas 20 May 2005 Overview Histogram Matching This example will take four Digital Ortho Quads that have different ...
