The chi-square test asks whether observed counts differ from what you would expect by chance. Enter a contingency table to test whether two categorical variables are independent, or supply expected frequencies for a goodness-of-fit test. Expected counts are calculated for you and the assumptions are checked.
The test of independence asks whether two categorical variables are related - does treatment group predict outcome, does region predict preference. Expected counts come from the row and column totals, on the assumption that the variables are unrelated.
The goodness-of-fit test asks whether one set of counts matches a distribution you specify. Leave the expected counts blank and it tests against a uniform distribution, which is what you want for checking whether a die is fair.
Chi-square is an approximation that relies on reasonably large expected counts. The usual guidance is that no expected count should fall below 1, and no more than 20% should fall below 5. When your table breaches that, the p-value is not trustworthy and Fisher's exact test is the right alternative. The calculator flags it when this happens.
Chi-square grows with sample size, so with enough data a negligible association becomes "significant". Cramer's V rescales the result to a 0-1 range that does not depend on n, which makes it the more informative number to report.
For a contingency table it is (rows - 1) x (columns - 1), so a 2x2 table has 1 degree of freedom. For a goodness-of-fit test it is the number of categories minus 1, minus one more for each parameter you estimated from the data.
The chi-square approximation becomes unreliable. The usual rule is that all expected counts should be at least 1 and no more than 20% should be below 5. If yours fail that, use Fisher's exact test instead. The calculator warns you when this happens.
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