Noncentral t-Distribution Calculator

Compute the PDF, lower-tail CDF, upper-tail probability, mean, and variance of the noncentral t-distribution for a given degrees of freedom and noncentrality, with charts.

Input

Enter the statistic t, the degrees of freedom v, and the noncentrality d to compute the density, cumulative distribution (lower-tail probability), upper-tail probability, mean, and variance of the noncentral t-distribution.

Statistic t

The t value to evaluate. Negative values are allowed.

Degrees of freedom v

Enter a value greater than 0.

Noncentrality d

The shift parameter of the distribution. A value of 0 gives the ordinary t-distribution.

Result

Lower-tail probability P(T at most t) for t = 2, v = 10, d = 1.5

0.6591541

Upper-tail probability P(T at least t)

0.3408459

Probability density f(t)

0.31460592

Mean

1.62558346

Variance

1.41997841

Probability density function (PDF)

Cumulative distribution function (CDF)

How it works

The noncentral t-distribution describes a standard normal variable with a nonzero mean divided by the square root of a scaled chi-square variable, and is used in power analysis and confidence intervals for effect sizes.
The cumulative distribution function CDF is computed with a Poisson-weighted series of regularized incomplete beta functions driven by the noncentrality delta.
The probability density PDF is obtained by central-difference numerical differentiation of the CDF.
The mean is defined only when the degrees of freedom exceed 1, and the variance only when they exceed 2.
When the noncentrality delta is zero, the distribution reduces to the ordinary central Student t-distribution.

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