Chi-square or Fisher's exact test?

Decided by your expected counts, not your observed ones. This trips people up, because the expected counts are not the numbers in your table.

The rule

Use Fisher's exact test when any expected cell count is below 5. Use chi-square otherwise.

Expected counts are what you would see if the two variables were unrelated, calculated from the row and column totals. Your software reports them. A table with an observed count of 2 may have perfectly acceptable expected counts.

Why the rule exists

Chi-square approximates a continuous distribution. With small expected counts that approximation breaks down and the p-value becomes unreliable. Fisher's test computes the probability exactly by enumeration, so it needs no approximation.

Practical guidance

Fisher's exact test is always valid for a 2x2 table, whatever the counts. With modern computing there is no real cost to using it by default for 2x2 tables, and doing so avoids the argument entirely.

For larger tables, chi-square is the practical choice, though exact tests are available for moderate sizes.

Yates's continuity correction

Applied by default to 2x2 chi-square tests in some software. It is widely considered overly conservative. If your counts are large enough for chi-square, you do not need it; if they are not, use Fisher's instead.

Paired data needs a different test

If the same subjects are measured twice, before and after, chi-square and Fisher's are both wrong because they assume independent observations. Use McNemar's test for 2x2 paired data, or Cochran's Q for more than two paired conditions.

Report an effect size

For 2x2 tables, the odds ratio or risk ratio with a confidence interval. A p-value alone tells your reader nothing about how large the association is.

In Plotara

Chi-square, Fisher's exact, McNemar, Cochran's Q, goodness of fit, binomial and negative binomial, with expected counts shown so the choice is visible.