Chi-Square Distribution Percent Point (Quantile)

Find the chi-square percent point (the χ² value) from a probability and degrees of freedom. Supports lower and upper tail probabilities for test critical values.

Input

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

Probability p

A value between 0 and 1 (for example 0.95)

Probability mode

Whether p is the lower tail or the upper tail probability

Degrees of freedom k

A positive value, non integers allowed (for example 5)

Result

Percent point chi-square for probability 0.95 and 5 degrees of freedom

11.07049769

Degrees of freedom k

Specified probability p

0.95

Lower tail CDF

0.95

Upper tail probability

0.05

Density at the point

0.01932464

Mean

Variance

Probability density function PDF

Cumulative distribution function CDF

How it works

The chi-square distribution equals a gamma distribution whose shape is half the degrees of freedom and whose scale is two, so its cumulative distribution function is the regularized lower incomplete gamma function.
The percent point is the inverse of the cumulative distribution function. This tool brackets the inverse regularized incomplete gamma with bisection, then refines it with Newton iteration using the probability density as the derivative.
In lower mode the input probability p is treated as the lower tail probability (CDF). In upper mode the value one minus p is converted to a lower tail probability before the inverse is solved.
The mean of the chi-square distribution is the degrees of freedom k and the variance is 2k. As the degrees of freedom grow the distribution approaches the normal distribution.
In a chi-square test the point whose upper tail probability equals the significance level alpha is the right side critical value. For example with 5 degrees of freedom and significance 0.05, choosing upper mode with p of 0.05 gives about 11.07.

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