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Noncentral Chi-Square Percent Point Calculator

Find the percent point (quantile) x of the noncentral chi-square distribution from a probability p, degrees of freedom k, and noncentrality lambda. Supports lower and upper tail modes, with density, mean, variance, and PDF/CDF charts.

Input

Enter a probability, mode, degrees of freedom, and noncentrality to find the percent point (quantile) of the noncentral chi-square distribution.

A value strictly between 0 and 1 (for example 0.95).

Mode

Choose whether p is treated as the lower tail or the upper tail probability.

A positive real number (for example 5).

A real number of 0 or greater. A value of 0 gives the ordinary chi-square distribution.

Result

Percent point x for p = 0.95, k = 5, λ = 4

18.62581506

Degrees of freedom k

5

Noncentrality λ

4

Probability p

0.95

Lower tail F(x)

0.95

Upper tail 1 − F(x)

0.05

Density f(x)

0.01302845

Mean

9

Variance

26

Probability density function (PDF)

Cumulative distribution function (CDF)

How it works

  • The noncentral chi-square distribution is a Poisson weighted mixture of central chi-square distributions. Its CDF is F(x)=sum of e^(-lambda/2)(lambda/2)^j/j! times P((k+2j)/2, x/2), where P is the regularized lower incomplete gamma function.
  • The percent point x satisfies F(x)=p in lower mode, or 1-F(x)=p in upper mode. The CDF is inverted by bracketing with bisection and refining with Newton iteration using the density as the derivative.
  • The mean is k+lambda and the variance is 2(k+2lambda). When the noncentrality lambda is 0 the distribution reduces to the ordinary central chi-square distribution.
  • Degrees of freedom k must be a positive real number and noncentrality lambda must be 0 or greater. The probability p must lie strictly between 0 and 1.

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