Chi-Square Distribution Calculator

Result

What this answers

This calculator gives direct access to the chi-square distribution, the distribution behind the Chi-Square Test of Independence, Levene's test, and variance-based methods elsewhere on this site. It answers "what fraction of this distribution falls at or below (or at or above) a specific value?"

How it is calculated

The chi-square cumulative distribution function is computed via its exact relationship to the regularized lower incomplete gamma function, the same core numerical routine this project's log-gamma and incomplete-gamma implementation provides for every other engine that needs a chi-square value.

Worked example

For 1 degree of freedom and x=3.841458821 (a widely published critical value): the upper-tail probability works out to almost exactly .05, confirming this is the standard alpha=.05 critical value for a chi-square test with 1 degree of freedom.

Assumption audit

Calculated from your data: the exact cumulative or upper-tail probability for your stated degrees of freedom and value.

Evidence to review: whether your degrees of freedom actually match the test you are cross-checking; degrees of freedom differ by test (rows minus 1 times columns minus 1 for independence, groups minus 1 for variance tests, and so on).

You must verify: that a chi-square-based method is actually the right choice for your data in the first place; this tool looks up distribution values only.

Common mistakes

A common mistake is entering a negative x value, which is impossible for the chi-square distribution (it is defined only for nonnegative values) and correctly throws an error here. Another is confusing this raw lookup with an actual completed chi-square test; running the Chi-Square Test of Independence on your own data computes both the statistic and its p-value together, rather than requiring you to look up the p-value separately. A third is using the wrong degrees of freedom for the specific chi-square-based method being cross-checked, since the formula for degrees of freedom differs by test.

What this result does not mean

A small upper-tail probability from this lookup only means the observed chi-square value would be rare under the assumed model at this specific degrees of freedom; it says nothing about which specific comparisons drove that value, nor about practical importance, which this raw distribution lookup cannot address on its own.

Limitations

This is a raw distribution lookup tool; it does not compute a chi-square statistic from your own data or verify that a chi-square-based method's assumptions actually apply.