In statistics, a histogram is a technique to estimate the probability distribution of a variable, by counting the frequencies of data into discrete bins, and then plotting the number of members in each bin versus the bin number.1 This is usually but not necessarily displayed by a bar chart where each bar is erected over an interval, 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
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.3
Contents 1 Etymology 2 Examples 3 Activities and demonstrations 4 Mathematical definition 4.1 Cumulative histogram 4.2 Number of bins and width 5 See also 6 References 7 Further reading 8 External links // Etymology Look up histogram in Wiktionary, the free dictionary. 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 1895, derived the name from "historical diagram".
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ExamplesAs an example we consider data collected by the U.S. Census Bureau on time to travel to work (2000 census, [1], 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 15 but less than 20 minutes" is higher than for the bands on either side. This is likely to have arisen from people rounding their reported journey time.original research? 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 57This 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 0.0221 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 0.0005This 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.)5
Activities and demonstrationsThe 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 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 mi 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 mi meets the following conditions:
Cumulative histogramA 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 mj is defined as:
Number of bins and widthThere 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. 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.6
The number of bins k can be assigned directly or can be calculated from a suggested bin width h as:
The braces indicate the ceiling function.
Sturges' formula7which implicitly bases the bin sizes on the range of the data, and can perform poorly if n < 30.
Scott's choice8where σ is the sample standard deviation.
Square-root choicewhich takes the square root of the number of data points in the sample (used by Excel histograms and many others)
Freedman–Diaconis' choice9which is based on the interquartile range.
Choice based on minimization of an estimated L2 risk function10where and are mean and biased variance of a histogram with bin-width , and .
See also Wikimedia Commons has media related to: Histograms Look up histogram in Wiktionary, the free dictionary. Statistics portal Data binning Freedman–Diaconis rule Image histogram Density estimation Kernel density estimation, a smoother but more complex method of density estimation References ^ "Histogram". Wolfram Research. http://mathworld.wolfram.com/Histogram.html. ^ Howitt, D. and Cramer, D. (2008) Statistics in Psychology. Prentice Hall ^ Nancy R. Tague (2004). "Seven Basic Quality Tools". The Quality Toolbox. Milwaukee, Wisconsin: American Society for Quality. p. 15. http://www.asq.org/learn-about-quality/seven-basic-quality-tools/overview/overview.html. Retrieved 2010-02-05. ^ 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 volume. ISSN 1177–1380. http://www.rutherfordjournal.org/article010107.html. ^ Dean, S., & Illowsky, B. (2009, February 19). Descriptive Statistics: Histogram. Retrieved from the Connexions Web site: http://cnx.org/content/m16298/1.11/ ^ e.g. § 5.6 "Density Estimation", W. N. Venables and B. D. Ripley, Modern Applied Statistics with S, Springer, 4th edition ^ 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: 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. http://www.mitpressjournals.org/doi/abs/10.1162/neco.2007.19.6.1503. Further reading 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: 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 v • d • e 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 Grouped data · Frequency distribution · Contingency table Dependence Pearson product-moment correlation · Rank correlation (Spearman's rho, Kendall's tau) · Partial correlation · Scatter plot Statistical graphics Bar chart · Biplot · Box plot · Control chart · Correlogram · Forest plot · Histogram · Q-Q plot · Run chart · Scatter plot · Stemplot · Radar chart 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 · Regression discontinuity · Optimal design Uncontrolled studies Natural experiment · Quasi-experiment · Observational study Statistical inference Bayesian inference Prior · Posterior · Credible interval · Bayes factor · Bayesian estimator · Maximum posterior estimator Classical inference Confidence interval · Hypothesis testing · Sampling distribution · Meta-analysis Specific tests Z-test (normal) · Student's t-test · F-test · Chi-square test · Pearson's chi-square · Wald test · Mann–Whitney U · Shapiro–Wilk · Signed-rank General estimation Mean-unbiased · Median-unbiased · Maximum likelihood · Method of moments · Minimum distance · Maximum spacing · Density estimation Correlation and regression analysis Correlation Pearson product-moment correlation · Partial correlation · Confounding variable · Coefficient of determination Linear regression Simple linear regression · Ordinary least squares · General linear model · Analysis of variance · Analysis of covariance Non-standard predictors Nonlinear regression · Nonparametric · Semiparametric · Isotonic · Robust Generalized linear model Exponential families · Logistic (Bernoulli) · Binomial · Poisson Data analyses and models for other specific data types Multivariate statistics Multivariate regression · Principal components · Factor analysis · Cluster analysis · Copulas Time series analysis Decomposition · Trend estimation · Box–Jenkins · ARMA models · Spectral density estimation Survival analysis Survival function · Kaplan–Meier · Logrank test · Failure rate · Proportional hazards models · Accelerated failure time model Categorical data McNemar's test · Cohen's kappa Applications Environmental statistics Geostatistics · Climatology Medical statistics Epidemiology · Clinical trial · Clinical study design Social statistics Actuarial science · Population · Demography · Census · Psychometrics · Official statistics · Crime statistics Category · Portal · Outline · Indexhistogram ( ) n. A bar graph of a frequency distribution in which the widths of the bars are proportional to the classes into which the variable has
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12 5 = 2 4 Significance of area The area on a histogram is important in being able to find the total number of values individual results in the data In our histogram from the table the
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Recovery of the histogram of hourly ozone distribution from weekly average concentrations [An article from: Environmental Pollution] Elsevier
The histogram is an important tool for exploratory data analysis...
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The full-featured Nikon Coolpix P7000 is geared to go head-to-head with Canon's G series of point-and-shoot cameras, offering fast access to manual controls and performance-minded hardware. Nikon - Camera - Point-and-shoot camera - Nikon Coolpix series - Photography
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The purpose of a histogram (Chambers) is to graphically summarize the distribution of a univariate data set. The histogram graphically shows the following: ...
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pie graph 2 the distribution of separate values of a variable in relation to another scatter graph or 3 change in the value of a variable in relation to another coordinate graph histogram etc for example change in the average price of a journal subscription over time
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Some insights about college students' interpretations of histograms.: An article from: Focus on Learning Problems in Mathematics Thomson Gale
How to Create a Histogram Using the Frequency Function in Excel. Part of a Monte Carlo Simulation Example.
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The full-featured Nikon Coolpix P7000 is geared to go head-to-head with Canon's G series, offering fast access to manual controls and performance-minded hardware.
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Pine Barrens photography: Reading a Histogram for correct exposure
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Histograms are the key to understanding digital images. This 10x4 mosaic contains 40 tiles which we could sort by color and then stack up accordingly. ...
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Nikon made the Coolpix P7000 a much different camera from its predecessor, the gone-nowhere P6000, probably in hope of a better showing against market-leading models like the Canon PowerShot G11 and Panasonic Lumix DMC-LX3.
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The histogram is only appropriate for variables whose values are numerical and measured on an interval scale. It is generally used when dealing ...
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Years after Canon first revealed its Powershot G-series, Nikon is finally giving their loyalist customers a pro-level point-and-shoot of their very own in the Nikon P7000. The P7000 is clearly inspired by the Canon G-Series in its build quality, resembling a G11 in almost every detail except its branding. Intelligently, on the inside, Nikon has opted [...]
of classes bars or have DreamCalc determine a suitable number using Scott s Rule by selecting Auto from the drop down list An example histogram and cumulative variation are shown below Histogram Chart
Histogram video 2
Histogram: Webster's Timeline History, 1951 - 2007 ICON Group International, Inc.
Histograms are one of the most important and valuable tools we have when editing, retouching or restoring images in Photoshop, and knowing how to ...
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The Nikon Coolpix P7000 could be a Canon PowerShot and Panasonic Lumix killer. Read the full story on the Nikon Coolpix P7000 here.
Print This is a histogram with the nodes arranged horizontally They are Active charts which differ from the normal histograms in that if you click the chart you can find out which records are linked to each element
Histogram video 1
Leader manual and instructional guide: Histograms Quality Circle Institute
A histogram is the graphical version of a table which shows what proportion of cases fall into each of several or many specified categories. ...
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Nikon has outed three new cameras today, and first up is the Nikon COOLPIX P7000 . The new flagship to the company’s COOLPIX range, the P7000 packs a 10.1-megapixel CCD sensor and 7.1x optical zoom into a consumer-sized chassis, complete with the sort of controls you might normally expect to find on a DSLR. That allows for plenty of manual tweaking, useful for those who want some of the ...
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Do you like this story? The Apple iPad is a pretty well-rounded product, but it’s missing some rather important features. What if you need to read an SD card? What if you need extra storage space?
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Well well -- what's this? Look like all those rumors about Nikon licking its wounds and gunning hard for the high-end compact camera market were true: this hot little piece is the new Coolpix P7000, Nikon's answer to the Canon G11. Like we'd heard, it actually drops the megapixel count from the P6000, with a 10.1 megapixel CCD sensor that can hit ISO 6400 sensitivity natively and 12,800 when ...
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Flow around a square cylinder (Re=10000) [with Ehrenfests' histogram ...