Chisquare {stats}R Documentation

The (non-central) Chi-Squared Distribution


Density, distribution function, quantile function and random generation for the chi-squared (chi^2) distribution with df degrees of freedom and optional non-centrality parameter ncp.


dchisq(x, df, ncp=0, log = FALSE)
pchisq(q, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
qchisq(p, df, ncp=0, lower.tail = TRUE, log.p = FALSE)
rchisq(n, df, ncp=0)


x, q vector of quantiles.
p vector of probabilities.
n number of observations. If length(n) > 1, the length is taken to be the number required.
df degrees of freedom (non-negative, but can be non-integer).
ncp non-centrality parameter (non-negative).
log, log.p logical; if TRUE, probabilities p are given as log(p).
lower.tail logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].


The chi-squared distribution with df= n > 0 degrees of freedom has density

f_n(x) = 1 / (2^(n/2) Gamma(n/2)) x^(n/2-1) e^(-x/2)

for x > 0. The mean and variance are n and 2n.

The non-central chi-squared distribution with df= n degrees of freedom and non-centrality parameter ncp = λ has density

f(x) = exp(-lambda/2) SUM_{r=0}^infty ((lambda/2)^r / r!) dchisq(x, df + 2r)

for x >= 0. For integer n, this is the distribution of the sum of squares of n normals each with variance one, λ being the sum of squares of the normal means; further,
E(X) = n + λ, Var(X) = 2(n + 2*λ), and E((X - E(X))^3) = 8(n + 3*λ).

Note that the degrees of freedom df= n, can be non-integer, and for non-centrality λ > 0, even n = 0; see Johnson et al. (1995, chapter 29).

Note that ncp values larger than about 1e5 may give inaccurate results with many warnings for pchisq and qchisq.


dchisq gives the density, pchisq gives the distribution function, qchisq gives the quantile function, and rchisq generates random deviates.
Invalid arguments will result in return value NaN, with a warning.


The central cases are computed via the gamma distribution.

The non-central dchisq and rchisq are computed as a Poisson mixture central of chi-squares (Johnson et al, 1995, p.436).

The non-central pchisq is for ncp < 80 computed from the Poisson mixture of central chi-squares and for larger ncp based on a C translation of

Ding, C. G. (1992) Algorithm AS275: Computing the non-central chi-squared distribution function. Appl.Statist., 41 478–482.

which computes the lower tail only (so the upper tail suffers from cancellation).

The non-central qchisq is based on inversion of pchisq.


Becker, R. A., Chambers, J. M. and Wilks, A. R. (1988) The New S Language. Wadsworth & Brooks/Cole.

Johnson, N. L., Kotz, S. and Balakrishnan, N. (1995) Continuous Univariate Distributions, chapters 18 (volume 1) and 29 (volume 2). Wiley, New York.

See Also

A central chi-squared distribution with n degrees of freedom is the same as a Gamma distribution with shape a = n/2 and scale s = 2. Hence, see dgamma for the Gamma distribution.


dchisq(1, df=1:3)
pchisq(1, df= 3)
pchisq(1, df= 3, ncp = 0:4)# includes the above

x <- 1:10
## Chi-squared(df = 2) is a special exponential distribution
all.equal(dchisq(x, df=2), dexp(x, 1/2))
all.equal(pchisq(x, df=2), pexp(x, 1/2))

## non-central RNG -- df=0 is ok for ncp > 0:  Z0 has point mass at 0!
Z0 <- rchisq(100, df = 0, ncp = 2.)

## Not run: 
## visual testing
## do P-P plots for 1000 points at various degrees of freedom
L <- 1.2; n <- 1000; pp <- ppoints(n)
op <- par(mfrow = c(3,3), mar= c(3,3,1,1)+.1, mgp= c(1.5,.6,0),
          oma = c(0,0,3,0))
for(df in 2^(4*rnorm(9))) {
  plot(pp, sort(pchisq(rr <- rchisq(n,df=df, ncp=L), df=df, ncp=L)),
       ylab="pchisq(rchisq(.),.)", pch=".")
  mtext(paste("df = ",formatC(df, digits = 4)), line= -2, adj=0.05)
mtext(expression("P-P plots : Noncentral  "*
                 chi^2 *"(n=1000, df=X, ncp= 1.2)"),
      cex = 1.5, font = 2, outer=TRUE)
## End(Not run)

[Package stats version 2.5.0 Index]