represents an F-ratio distribution with n numerator and m denominator degrees of freedom.


  • FRatioDistribution is also known as the FisherSnedecor distribution.
  • The probability density for value in an F-ratio distribution is proportional to for , and is zero for . »
  • For integers n and m, the F-ratio distribution gives the distribution of the ratio of variances for samples from normal distributions.
  • FRatioDistribution allows n and m to be any positive real numbers.
  • FRatioDistribution allows n and m to be dimensionless quantities. »
  • FRatioDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

Background & Context

  • FRatioDistribution[n,m] represents a continuous statistical distribution over the interval defined as the distribution of the ratio X=Y1/Y2, where Y1ChiSquareDistribution[n] and Y2ChiSquareDistribution[m] are independent variates with n and m degrees of freedom, respectively. Depending on the values of n and m, the probability density function (PDF) may be either unimodal or monotonically decreasing with a potential singularity nearing the lower endpoint of its domain. In addition, the tails of the PDF are "fat" in the sense that the PDF decreases algebraically rather than exponentially for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) The F-ratio distribution is sometimes called the FisherSnedecor distribution, (central) F-distribution, or Snedecor's F-distribution.
  • The F-ratio distribution was first formalized in the mid-1930s by American mathematician G. W. Snedecor as a tool to improve the analysis of variance as introduced by English statistician R. A. Fisher in the late 1910s. The F-ratio distribution is a staple in modern statistics, where it forms the basis for the so-called F-test. The F-test is a test whose null hypothesis involves an F-ratio distributed test statistic. It is used to test a number of common hypotheses, examples of which include determining if a proposed regression model fits a certain dataset and if the means of normally distributed populations with the same standard deviation are equal. Because of the generality of such tests, the F-ratio distribution is used in many fields, including economics, psychology, pharmacy, engineering, medicine, and manufacturing.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from an F-ratio distribution. Distributed[x,FRatioDistribution[n,m]], written more concisely as xFRatioDistribution[n,m], can be used to assert that a random variable x is distributed according to an F-ratio distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions may be given using PDF[FRatioDistribution[n,m],x] and CDF[FRatioDistribution[n,m],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively.
  • DistributionFitTest can be used to test if a given dataset is consistent with an F-ratio distribution, EstimatedDistribution to estimate an F-ratio parametric distribution from given data, and FindDistributionParameters to fit data to an F-ratio distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic F-ratio distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic F-ratio distribution.
  • TransformedDistribution can be used to represent a transformed F-ratio distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain an F-ratio distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving F-ratio distributions.
  • The F-ratio distribution is related to a number of other distributions. In particular, as previously noted, FRatioDistribution is defined in terms of the ChiSquareDistribution (and hence related to ChiDistribution). Moreover, the PDF of FRatioDistribution[1,m] tends to that of ChiSquareDistribution[1] as m tends to Infinity. FRatioDistribution can be obtained from other distributions both by simple transformations, e.g. as the square of StudentTDistribution, as well as from more complicated transformations of distributions such as LaplaceDistribution and FisherZDistribution. FRatioDistribution is a special case of NoncentralFRatioDistribution, in the sense that the PDF of FRatioDistribution[n,m] is precisely that of both NoncentralFRatioDistribution[n,m,0] and NoncentralFRatioDistribution[n,m,0,0] and is also related to BetaDistribution, PearsonDistribution, HotellingTSquareDistribution, and BinomialDistribution.


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Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:


Scope  (8)

Generate a sample of pseudorandom numbers from an F-ratio distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the density histogram of the sample with the PDF of the estimated distribution:

Skewness varies with the degrees of freedom:

Limiting value:

Kurtosis varies with the degrees of freedom:

Limiting value is the kurtosis of NormalDistribution:

Different moments with closed forms as functions of parameters:


Closed form for symbolic order:




Hazard function:

Quantile function:

Use dimensionless Quantity to specify the degree of freedom parameters n and m:

Applications  (1)

FRatioDistribution is the distribution of the ratio of two sample variances drawn from two normal distributions. Define the Fisher ratio statistic:

Generate 1000 batches of samples from two standard normal distributions:

Compute values of the Fisher ratio statistics for each batch:

Find the value of the Fisher ratio test for the first batch:

Compare with FisherRatioTest:

Properties & Relations  (13)

FRatioDistribution is closed under inverse:

Relationships to other distributions:

ChiSquareDistribution is a limiting case of F-ratio distribution:

F-ratio is the ratio of two ChiSquareDistribution variables:

F-ratio distribution can be obtained from BetaDistribution:

A square of StudentTDistribution has F-ratio distribution:

F-ratio distribution is the distribution of the inverse square of StudentTDistribution:

F-ratio distribution is a special case of BetaPrimeDistribution:

F-ratio distribution is a special case of type 6 PearsonDistribution:

FRatioDistribution is a transformation of Laplace distribution:

FisherZDistribution is a transformation of F-ratio distribution:

NoncentralFRatioDistribution simplifies to F-ratio distribution:

Doubly NoncentralFRatioDistribution simplifies to F-ratio distribution:

Possible Issues  (2)

FRatioDistribution is not defined when n or m is not a positive real number:

Substitution of invalid parameters into symbolic outputs gives results that are not meaningful:

Neat Examples  (1)

PDFs for different n values with CDF contours:

Wolfram Research (2007), FRatioDistribution, Wolfram Language function, (updated 2016).


Wolfram Research (2007), FRatioDistribution, Wolfram Language function, (updated 2016).


Wolfram Language. 2007. "FRatioDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016.


Wolfram Language. (2007). FRatioDistribution. Wolfram Language & System Documentation Center. Retrieved from


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@online{reference.wolfram_2024_fratiodistribution, organization={Wolfram Research}, title={FRatioDistribution}, year={2016}, url={}, note=[Accessed: 24-May-2024 ]}