represents a noncentral Student distribution with ν degrees of freedom and noncentrality parameter δ.


Background & Context

  • NoncentralStudentTDistribution[ν,δ] represents a continuous statistical distribution defined and supported over the set of real numbers and parametrized by a real number δ (called a "noncentrality parameter") and by a positive real number ν (the "degrees of freedom") that together determine the overall behavior of its probability density function (PDF). In general, the PDF of a noncentral Student -distribution is unimodal with a single "peak" (i.e. a global maximum), though its overall shape (its height, its spread, and the horizontal location of its maximum) is determined by the values of ν and δ. 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.) NoncentralStudentTDistribution is a perhaps-skewed generalization of the Student -distribution (StudentTDistribution, sometimes referred to as the centralized Student -distribution) and is sometimes referred to as the noncentral -distribution.
  • The noncentral -distribution was first devised in the 1930s, though its first thorough quantitative analysis came in a 1940 article by Johnson and Welch. Originally, the distribution was realized as the distribution describing the behavior of a random variate of the form where zNormalDistribution[], where wChiSquareDistribution[ν]/ν, and where δ is the noncentrality parameter. In the years since, the noncentral -distribution has found a number of fundamental uses throughout statistics such as Bayesian inference, normality testing, and modeling the power of null hypothesis test for a population chosen according to a (central) -distribution. The distribution is also used in quality control, microbiology, macroeconomics, and pharmaceuticals.
  • RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a noncentral -distribution. Distributed[x,NoncentralStudentTDistribution[ν,δ]], written more concisely as xNoncentralStudentTDistribution[ν,δ], can be used to assert that a random variable x is distributed according to a noncentral -distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation.
  • The probability density and cumulative distribution functions for noncentral -distributions may be given using PDF[NoncentralStudentTDistribution[ν,δ],x] and CDF[NoncentralStudentTDistribution[ν,δ],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 a noncentral -distribution, EstimatedDistribution to estimate a noncentral parametric distribution from given data, and FindDistributionParameters to fit data to a noncentral -distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic noncentral -distribution, and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic noncentral -distribution.
  • TransformedDistribution can be used to represent a transformed noncentral -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 a noncentral -distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving noncentral -distributions.
  • NoncentralStudentTDistribution is related to a number of other distributions. It is an immediate generalization of StudentTDistribution in the sense that the PDF of NoncentralStudentTDistribution[ν,0] is precisely that of StudentTDistribution[ν]. NoncentralStudentTDistribution is also related to NormalDistribution, ChiSquareDistribution, PearsonDistribution, MultivariateTDistribution, FRatioDistribution, GammaDistribution, and HotellingTSquareDistribution.


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Basic Examples  (3)

Probability density function:

Cumulative distribution function:

Mean and variance:

Scope  (7)

Generate a sample of pseudorandom numbers from a noncentral Student 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 is defined for :

Kurtosis is defined for :

Different moments with closed forms as functions of parameters:


Closed form for symbolic order:




Hazard function has a strong mode for negative values of parameter δ:

Quantile function:

Applications  (1)

The weight, in grams, of a particular boxed cereal product is known to follow a normal distribution with unknown mean . A test is performed with null hypothesis and alternative hypothesis . Fifteen boxes were chosen at random with sample mean weight of 363 and standard deviation of 32:

Critical value of statistic at the 5% level:

Hence, the -test does not reject the null hypothesis:

Compute the power of the test, given and . In this case the test statistic follows a NoncentralStudentTDistribution[n-1,δ]:

The power of the -test to reject the null hypothesis is low:

Plot the power of the test as a function of sample size:

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

Properties & Relations  (3)

Relationships to other distributions:

NoncentralStudentTDistribution[ν,0] is equivalent to StudentTDistribution[ν]:

Noncentral distribution can be obtained from NormalDistribution and ChiSquareDistribution:

Possible Issues  (4)

NoncentralStudentTDistribution is not defined when ν is not a positive real number:

NoncentralStudentTDistribution is not defined when δ is not a real number:

The characteristic function of a noncentral Student distribution has no closed-form representation:

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

Neat Examples  (1)

PDFs for different δ values with CDF contours:

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


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


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


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


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@online{reference.wolfram_2024_noncentralstudenttdistribution, organization={Wolfram Research}, title={NoncentralStudentTDistribution}, year={2016}, url={}, note=[Accessed: 23-June-2024 ]}