represents a distribution with ν degrees of freedom.


  • The probability density for value in a distribution is proportional to for , and is zero for . »
  • For integer ν, the distribution with ν degrees of freedom gives the distribution of sums of squares of ν values independently sampled from a normal distribution.
  • ChiSquareDistribution allows ν to be any positive real number.
  • ChiSquareDistribution allows ν to be a dimensionless quantity. »
  • ChiSquareDistribution can be used with such functions as Mean, CDF, and RandomVariate. »

Background & Context


open allclose all

Basic Examples  (4)

Probability density function:

Cumulative distribution function:

Mean and variance:


Scope  (8)

Generate a sample of pseudorandom numbers from a distribution:

Compare its histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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


For a large number of degrees of freedom, the distribution becomes symmetric:


The limiting value is the kurtosis of NormalDistribution:

Different moments with closed forms as functions of parameters:


Closed form for symbolic order:


Closed form for symbolic order:



Cumulant has closed form:

Hazard function:

Quantile function:

Use dimensionless Quantity to specify the degree of freedom parameter ν:

Applications  (2)

ChiSquareDistribution is used in exact (small) sampling theory. Define statistics:

If data comes from a NormalDistribution, then statistics follow ChiSquareDistribution, even for data that is a sample of small size (less than 30):

The weight in grams of a particular boxed cereal product is known to follow a normal distribution. A quality assurance team samples 15 boxes at random and records their weights. Test the hypothesis that the standard deviation of the product weight is less than 36:

Under the null hypothesis of , the following statistic follows ChiSquareDistribution:

The null hypothesis cannot be rejected at the 5% level:

Assuming that the standard deviation of the product weight equals 32, compute the probability of rejecting the null hypothesis, also known as the power of the test, at the 5% level as a function of sample size:

Find the sample size required for the power of the test to be at least 80%:

Properties & Relations  (23)

ChiSquareDistribution[ν] converges to a normal distribution as ν->:

Sum of -distributed variables follows distribution:

Relationships to other distributions:

NoncentralChiSquareDistribution simplifies to distribution:

distribution is a limiting case of FRatioDistribution:

The ratio of two -distributed variables follows FRatioDistribution:

Sum of squares of variables from NormalDistribution follows distribution:

distribution is a special case of GammaDistribution:

Scaled distribution follows GammaDistribution:

The square root of a variable follows the ChiDistribution:

Square of RayleighDistribution with is a special case of distribution:

Square of MaxwellDistribution with is a special case of distribution:

distribution and InverseChiSquareDistribution have an inverse relationship:

distribution is a special case of type 3 PearsonDistribution:

A transformation of distribution yields BetaDistribution:

is a transformation of UniformDistribution:

distribution is a transformation of LaplaceDistribution:

For sum of variables:

distribution is a transformation of ParetoDistribution:

distribution is a transformation of ParetoDistribution:

StudentTDistribution is a transformation of distribution:

StudentTDistribution can be obtained from ChiSquareDistribution and NormalDistribution:

NoncentralBetaDistribution can be obtained as a transformation of ChiSquareDistribution and NoncentralChiSquareDistribution:

NoncentralStudentTDistribution can be obtained from NormalDistribution and ChiSquareDistribution:

Possible Issues  (2)

ChiSquareDistribution is not defined when ν 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 ν values with CDF contours:

Introduced in 2007
Updated in 2016