Chi-Square Distribution Calculator
Look up the mean and variance of a Chi-Square (χ²) distribution from its degrees of freedom, for goodness-of-fit and independence tests.
Chi-Square Distribution (χ²)
Goodness-of-fit tests and test of independence in contingency tables.
Chi-Square (χ²) Distribution Mathematics
Chi-square distribution calculator computes the mean and variance of a χ² distribution from its degrees of freedom, for goodness-of-fit and independence tests.
The Chi-Square distribution is defined entirely by its degrees of freedom (df). Unlike the symmetric normal distribution, it is right-skewed and only takes non-negative values, which is why it's used to test whether observed categorical data matches an expected distribution.
- 1Enter the degrees of freedom (df) for the test and the observed chi-square statistic (χ²).
- 2View the distribution's mean and variance to understand where your statistic falls relative to the expected spread.
A categorical goodness-of-fit test has 5 degrees of freedom and produces a chi-square statistic of 11.07 (the standard 5% critical value for df=5).
Frequently Asked Questions (FAQ)
It tests whether observed categorical frequencies significantly differ from expected theoretical frequencies — common uses include goodness-of-fit tests and tests of independence between two categorical variables in a contingency table.
As degrees of freedom increase, the chi-square distribution becomes less skewed and starts to resemble a normal distribution — both its mean (= df) and variance (= 2×df) grow directly with degrees of freedom.