Chi-Square (χ²) Distribution Calculator

What is the chi-square distribution?

The chi-square (χ²) distribution is a right-skewed curve defined only for values ≥ 0. It is the distribution of a sum of squared standard normal variables, and its shape depends on the degrees of freedom. It underlies the goodness-of-fit test, the tests of independence and homogeneity for contingency tables, and tests about a population variance. As df increases the curve becomes more symmetric and approaches the normal distribution.

How to use this calculator

• χ² → P mode: Enter your χ² statistic and df to get the exact corresponding p-value. The shaded area on the graph equals the p-value.
• P → χ² mode: Enter a significance level (α, e.g. 0.05) and df to find the critical χ² value(s).
• Right-tailed is by far the most common: goodness-of-fit and independence tests reject H₀ only for large χ². Use left- or two-tailed for variance tests where a direction (or both) is of interest.