Histogram

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

Histogram

First described by	Karl Pearson
Purpose	To roughly assess the probability distribution of a given variable by depicting the frequencies of observations occurring in certain ranges of values

In statistics, a histogram is a graphical representation of the distribution of data. It is an estimate of the probability distribution of a continuous variable and was first introduced by Karl Pearson.^[1] A histogram is a representation of tabulated frequencies, shown as adjacent rectangles, erected over discrete intervals (bins), with an area equal to the frequency of the observations in the interval. The height of a rectangle is also equal to the frequency density of the interval, i.e., the frequency divided by the width of the interval. The total area of the histogram is equal to the number of data. A histogram may also be normalized displaying relative frequencies. It then shows the proportion of cases that fall into each of several categories, with the total area equaling 1. The categories are usually specified as consecutive, non-overlapping intervals of a variable. The categories (intervals) must be adjacent, and often are chosen to be of the same size.^[2] The rectangles of a histogram are drawn so that they touch each other to indicate that the original variable is continuous.^[3]

Histograms are used to plot the 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.^[4]

1 Etymology
2 Examples
3 Mathematical definition
- 3.1 Cumulative histogram
- 3.2 Number of bins and width
4 See also
5 References
6 Further reading
7 External links

Etymology [edit]

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

The etymology of the word histogram is uncertain. Sometimes it is said to be 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'. It is also said that Karl Pearson, who introduced the term in 1891, derived the name from "historical diagram".^[5]

Examples [edit]

The U.S. Census Bureau found that there were 124 million people who work outside of their homes.^[6] Using their data on the time occupied by travel to work, Table 2 below shows the absolute number of people who responded with travel times "at least 15 but less than 20 minutes" is higher than the numbers for the categories above and below it. This is likely due to people rounding their reported journey time.^{[citation needed]} The problem of reporting values as somewhat arbitrarily rounded numbers is a common phenomenon when collecting data from people.^{[citation needed]}

Histogram of travel time (to work), 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, with Q in thousands.

Histogram of travel time (to work), 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	0.0221
10	5	18618	0.0300
15	5	19634	0.0316
20	5	17981	0.0290
25	5	7190	0.0116
30	5	16369	0.0264
35	5	3212	0.0052
40	5	4122	0.0066
45	15	9200	0.0049
60	30	6461	0.0017
90	60	3435	0.0005

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. The intervals are placed together in order to show that the data represented by the histogram, while exclusive, is also continuous. (E.g., in a histogram it is possible to have two connecting intervals of 10.5–20.5 and 20.5–33.5, but not two connecting intervals of 10.5–20.5 and 22.5–32.5. Empty intervals are represented as empty and not skipped.)^[7]

Mathematical definition [edit]

An ordinary and a cumulative histogram of the same data. The data shown is a random sample of 10,000 points from a normal distribution with a mean of 0 and a standard deviation of 1.

In a more general mathematical sense, a histogram is a function m_i that counts the number of observations that fall into each of the 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:

$n = \sum_{i=1}^k{m_i}.$

Cumulative histogram [edit]

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_j is defined as:

$M_i = \sum_{j=1}^i{m_j}.$

Number of bins and width [edit]

There is no "best" number of bins, and different bin sizes can reveal different features of the data. Grouping data is as least as old as Graunt's work in the 17th century, but no systematic guidelines were given^[8] until Sturges's work in 1926^[9]

Some theoreticians have attempted to determine an optimal number of bins, but these methods generally make strong assumptions about the shape of the distribution. Depending on the actual data distribution and the goals of the analysis, different bin widths may be appropriate, so experimentation is usually needed to determine an appropriate width. There are, however, various useful guidelines and rules of thumb.^[10]

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

$k = \left \lceil \frac{\max x - \min x}{h} \right \rceil.$

The braces indicate the ceiling function.

Square-root choice

$k = \sqrt{n}, \,$

which takes the square root of the number of data points in the sample (used by Excel histograms and many others).^{[citation needed]}

Sturges' formula

Sturges' formula^[9] is derived from a binomial distribution and implicitly assumes an approximately normal distribution.

$k = \lceil \log_2 n + 1 \rceil, \,$

It implicitly bases the bin sizes on the range of the data and can perform poorly if n < 30.^{[citation needed]} It may also perform poorly if the data are not normally distributed.

Rice Rule

$h = 2 n^{1/3},$

The Rice Rule ^[11] is presented as a simple alternative to Sturges's rule.

Doane's formula

Doane's formula^[12] is a modification of Sturges' formula which attempts to improve its performance with non-normal data.

$k = 1 + log_2( n ) + log_2 ( 1 + \frac { |g_1| }{\sigma_{g_1}} )$

where $g_1$ is the estimated 3rd-moment-skewness of the distribution and

$\sigma_{g_1} = \sqrt { \frac { 6(n-2) }{ (n+1)(n+3) } }$

Scott's normal reference rule

$h = \frac{3.5 \hat \sigma}{n^{1/3}},$

where $\hat \sigma$ is the sample standard deviation. Scott's normal reference rule^[13] is optimal for random samples of normally distributed data, in the sense that it minimizes the integrated mean squared error of the density estimate.^[8]

Freedman–Diaconis' choice

The Freedman–Diaconis rule is:^[14]^[8]

$h = 2 \frac{\operatorname{IQR}(x)}{n^{1/3}},$

which is based on the interquartile range, denoted by IQR. It replaces 3.5σ of Scott's rule with 2 IQR, which is less sensitive than the standard deviation to outliers in data.

Choice based on minimization of an estimated L²^[15] risk function

$\underset{h}{\operatorname{arg\,min}} \frac{ 2 \bar{m} - v } {h^2}$

where $\textstyle \bar{m}$ and $\textstyle v$ are mean and biased variance of a histogram with bin-width $\textstyle h$ , $\textstyle \bar{m}=\frac{1}{k} \sum_{i=1}^{k} m_i$ and $\textstyle v= \frac{1}{k} \sum_{i=1}^{k} (m_i - \bar{m})^2$ .

Remark

A good reason why the number of bins should be proportional to $n^{1/3}$ is the following: suppose that the data are obtained as $n$ independent realizations of a bounded probability distribution with smooth density. Then the histogram remains equally »rugged« as $n$ tends to infinity. If $s$ is the »width« of the distribution (e. g., the standard deviation or the inter-quartile range), then the number of units in a bin (the frequency) is of order $n h/s$ and the relative standard error is of order $\sqrt{s/(n h)}$ . Comparing to the next bin, the relative change of the frequency is of order $h/s$ provided that the derivative of the density is non-zero. These two are of the same order if $h$ is of order $s/n^{1/3}$ , so that $k$ is of order $n^{1/3}$ .

This simple cubic root choice can also be applied to bins with non-constant width.

References [edit]

^ Pearson, K. (1895). "Contributions to the Mathematical Theory of Evolution. II. Skew Variation in Homogeneous Material". Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 186: 343–414. Bibcode:1895RSPTA.186..343P. doi:10.1098/rsta.1895.0010. edit
^ Howitt, D. and Cramer, D. (2008) Statistics in Psychology. Prentice Hall
^ Charles Stangor (2011) "Research Methods For The Behavioral Sciences". Wadsworth, Cengage Learning. ISBN 9780840031976.
^ Nancy R. Tague (2004). "Seven Basic Quality Tools". The Quality Toolbox. Milwaukee, Wisconsin: American Society for Quality. p. 15. Retrieved 2010-02-05.
^ M. Eileen Magnello (December 2006). "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 volume. OCLC 682200824.
^ US 2000 census.
^ Dean, S., & Illowsky, B. (2009, February 19). Descriptive Statistics: Histogram. Retrieved from the Connexions Web site: http://cnx.org/content/m16298/1.11/
^ ^a ^b ^c Scott, David W. (1992). Multivariate Density Estimation: Theory, Practice, and Visualization. New York: John Wiley.
^ ^a ^b Sturges, H. A. (1926). "The choice of a class interval". Journal of the American Statistical Association: 65–66. JSTOR 2965501.
^ e.g. § 5.6 "Density Estimation", W. N. Venables and B. D. Ripley, Modern Applied Statistics with S (2002), Springer, 4th edition. ISBN 0-387-95457-0.
^ Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University (chapter 2 "Graphing Distributions", section "Histograms")
^ Doane DP (1976) Aesthetic frequency classiﬁcation. American Statistician, 30: 181–183
^ 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.
^ Shimazaki, H.; Shinomoto, S. (2007). "A method for selecting the bin size of a time histogram". Neural Computation 19 (6): 1503–1527. doi:10.1162/neco.2007.19.6.1503. PMID 17444758.

External links [edit]

Look up histogram in Wiktionary, the free dictionary.

Journey To Work and Place Of Work (location of census document cited in example)
Smooth histogram for signals and images from a few samples
Histograms: Construction, Analysis and Understanding with external links and an application to particle Physics.
A Method for Selecting the Bin Size of a Histogram
Interactive histogram generator
Matlab function to plot nice histograms
Dynamic Histogram in MS Excel
Histogram construction and manipulation using Java applets, and charts on SOCR

Statistics

Descriptive statistics

Continuous data

Location	Mean (Arithmetic, Geometric, Harmonic) Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Designing studies	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling Stratified sampling Opinion poll Questionnaire

Controlled experiment	Design of experiments Randomized experiment Random assignment Replication Blocking Factorial experiment Optimal design

Uncontrolled studies	Natural experiment Quasi-experiment Observational study

Statistical inference

Statistical theory	Sampling distribution Order statistics Sufficiency Completeness Exponential family Permutation test (Randomization test) Empirical distribution Bootstrap U statistic Efficiency Asymptotics Robustness

Frequentist inference	Unbiased estimator (Mean unbiased minimum variance, Median unbiased) Biased estimators (Maximum likelihood, Method of moments, Minimum distance, Density estimation) Confidence interval Testing hypotheses Power Parametric tests (Likelihood-ratio, Wald, Score)

Specific tests	Z (normal) Student's t-test F Chi-squared Signed-rank (1-sample, 2-sample, 1-way anova) Shapiro–Wilk Kolmogorov–Smirnov

Bayesian inference	Bayesian probability Prior Posterior Credible interval Bayes factor Bayesian estimator Maximum posterior estimator

Correlation and regression analysis

Correlation	Pearson product–moment correlation Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust

Generalized linear model	Exponential families Logistic (Bernoulli) Binomial Poisson

Partition of variance	Analysis of variance (ANOVA) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical, multivariate, time-series, or survival analysis

Categorical data

Multivariate statistics

Time series analysis

General	Decomposition Trend Stationarity Seasonal adjustment

Time domain	ACF PACF XCF ARMA model ARIMA model Vector autoregression

Frequency domain	Spectral density estimation

Survival analysis

Applications

Biostatistics	Bioinformatics Clinical trials & studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process & Quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging