HistogramDistribution [{x1,x2,…}]
represents the probability distribution corresponding to a histogram of the data values xi.
HistogramDistribution [{{x1,y1,…},{x2,y2,…},…}]
represents a multivariate histogram distribution based on data values {xi,yi,…}.
HistogramDistribution […,bspec]
represents a histogram distribution with bins specified by bspec.
HistogramDistribution
HistogramDistribution [{x1,x2,…}]
represents the probability distribution corresponding to a histogram of the data values xi.
HistogramDistribution [{{x1,y1,…},{x2,y2,…},…}]
represents a multivariate histogram distribution based on data values {xi,yi,…}.
HistogramDistribution […,bspec]
represents a histogram distribution with bins specified by bspec.
Details
- HistogramDistribution returns a DataDistribution object that can be used like any other probability distribution.
- The probability density function for HistogramDistribution for a value is given by sum_(j=1)^m(c_j)/(n m_j) Boole[b_(j-1)<=x<b_j] where is the number of data points in bin , is the width of bin , are bin delimiters, and is the total number of data points.
- The width of each bin is computed according to the values xi, the width according to the yi, etc.
- The following bin specifications bspec can be given:
-
n use n bins{w} use bins of width w{min,max,w} use bins of width w from min to max{{b1,b2,…}} use bins [b1,b2),[b2,b3),…Automatic determine bin widths automatically"name" use a named binning methodfw apply fw to get an explicit bin specification {b1,b2,…}{xspec,yspec,…} give different x, y, etc. specifications
- Possible named binning methods include:
-
"FreedmanDiaconis" twice the interquartile range divided by the cube root of sample size"Knuth" balance likelihood and prior probability of a piecewise uniform model"Scott" asymptotically minimize the mean square error"Sturges" compute the number of bins based on the length of data"Wand" one-level recursive approximate Wand binning
- The probability density for value in a histogram distribution is a piecewise constant function.
- HistogramDistribution can be used with such functions as Mean , CDF , and RandomVariate .
Examples
open all close allBasic Examples (2)
Create a histogram distribution of univariate data:
Use the resulting distribution to perform analysis, including visualizing distribution functions:
Compute moments and quantiles:
Create a histogram distribution of bivariate data:
Visualize the PDF and CDF:
Compute covariance and general moments:
Scope (29)
Basic Uses (5)
Create a distribution from a histogram of some data:
Compute probabilities from the distribution:
Create histogram distributions from quantity data:
Find select descriptive statistics:
Decrease the number of bins to decrease local sensitivity:
Increase the bin width to decrease local sensitivity:
Create distributions from histograms in higher dimensions:
Plot the univariate marginal PDFs:
Plot the bivariate marginal PDFs:
Distribution Properties (10)
Estimate distribution functions:
Compute moments of the distribution:
Special moments:
General moments:
Quantile function:
Special quantile values:
Generate random numbers:
Compare with HistogramDistribution :
Compute probabilities and expectations:
Generating functions:
Estimate distribution functions for bivariate data:
Compute moments of a bivariate distribution:
Special moments:
General moments:
Generate random numbers:
Show the point distribution:
Having fewer bins yields a coarser approximation to the underlying distribution:
Generating functions:
Binning (14)
Automatically compute the number of bins:
More data yields smaller bins:
Explicitly specify the number of bins to use:
Specify 5 and 50 bins, respectively:
Explicitly specify bin width:
Use bin widths 1. and 0.1:
Specify bin range and bin width:
Use bin widths of 1.5 and .15 respectively over fixed interval:
Provide explicit bin delimiters:
Use different automatic binning methods:
Delimit bins on integer boundaries using a binning function:
Automatically compute the number of bins for bivariate data:
More data yields smaller bins:
Explicitly specify the number of bins to use:
Explicitly specify bin width:
Specify bin range and bin width:
Explicitly give bin delimiters:
Use different automatic binning methods:
Use different bin specifications in each dimension:
Specify 3 bins in the row dimension and bin width 0.5 in the column dimension:
Applications (6)
Compare an estimated density to a theoretical model:
Distribution of lengths of human chromosomes:
Compute the probability that the sequence length is greater than 15:
Compare the distributions of word length for some of the parts of speech:
The expected number of characters for a randomly chosen English noun:
Estimate the distribution of day-to-day point changes in the S&P 500 index:
Compute the probability of a 1% point change or more on a given day:
Determine the number of bins to use for bimodal data by Knuth's Bayesian method:
The optimal number of bins maximizes the log of the posterior density:
Density estimates using Knuth's method, Scott's rule, and the Freedman–Diaconis rule:
Knuth's method outperforms the other two in terms of LogLikelihood :
Construct a continuous version of the empirical cumulative distribution function:
Cumulative distribution function for HistogramDistribution is piecewise linear:
Compute Cramer–von Mises distance between the two distributions:
Properties & Relations (10)
The PDF of HistogramDistribution is equivalent to a probability density Histogram :
The resulting density estimate integrates to unity:
The precision of the output matches the precision of the data:
The PDF is piecewise constant:
The CDF and SurvivalFunction are piecewise linear:
The HazardFunction is linear fractional:
HistogramDistribution is a MixtureDistribution of uniform distributions:
HistogramDistribution is a consistent estimator of the underlying distribution:
HistogramDistribution works with the values only when the input is a TimeSeries :
Compare to the histogram distribution of the values:
HistogramDistribution works with all the values together when the input is a TemporalData :
Compare to the histogram distribution calculated with the values from all the paths:
Possible Issues (1)
It is possible to drop data from the estimation by specifying a binning range:
Specifying a width alone uses all the data:
Neat Examples (1)
Random pop art with HistogramDistribution :
Related Guides
Text
Wolfram Research (2010), HistogramDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/HistogramDistribution.html (updated 2016).
CMS
Wolfram Language. 2010. "HistogramDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/HistogramDistribution.html.
APA
Wolfram Language. (2010). HistogramDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/HistogramDistribution.html
BibTeX
@misc{reference.wolfram_2025_histogramdistribution, author="Wolfram Research", title="{HistogramDistribution}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/HistogramDistribution.html}", note=[Accessed: 16-November-2025]}
BibLaTeX
@online{reference.wolfram_2025_histogramdistribution, organization={Wolfram Research}, title={HistogramDistribution}, year={2016}, url={https://reference.wolfram.com/language/ref/HistogramDistribution.html}, note=[Accessed: 16-November-2025]}