Fisher Exact Test Calculator — Free 2×2 Analysis

Fisher's Exact Test. Eight passes against two failures looks very different from one pass against five failures. Fisher's exact test asks whether this sparse two-by-two imbalance is surprising. Enter four whole counts, labeled a, b, c, and d. Keep both row groups and both outcome columns visible because orientation controls interpretation. Fisher's null conditions on the observed row and column totals. Once those four margins are fixed, cell a alone determines every feasible table. Cell a ranges from the larger of zero and row one minus column two, through the smaller of row one and column one. For each feasible a, multiply two combination counts and divide by the total combination count. This is the conditional hypergeometric probability of that table. The two-sided rule sums every fixed-margin table no more probable than the observed table. It ranks table probabilities, not distance from an expected count. Compare the exact p-value with alpha for the decision. Report the sample odds ratio separately, because evidence strength and observed effect magnitude are different outputs.

Each independent observational unit must enter exactly one cell. The two variables must be genuinely binary, with mutually exclusive and consistently recorded categories. Exact arithmetic removes a large-sample approximation. It does not repair clustering, repeated measurements, selection bias, changing definitions, or outcome-driven stopping. Fisher is especially useful when expected cell counts are small and the chi-square approximation is weak. Healthy large tables usually give similar conclusions. The calculation conditions on the observed margins even when every margin was not fixed by design. That conditional frame is exact, discrete, and sometimes conservative. Distri Scope reports the sample odds ratio a times d over b times c. Swap one row or column and the ratio inverts. Do not confuse that sample ratio with R's conditional estimate. This page also does not provide an odds-ratio confidence interval, so precision remains unquantified. Paired binary outcomes need a paired test for binary outcomes. More than two categories need a larger-table method, and this page does not offer one-sided alternatives.

Suppose method one has eight passes and two failures, while method two has one pass and five failures. Plan a two-sided test at alpha point zero five. The row totals are ten and six; the column totals are nine and seven. Those margins allow a from three through nine, not four through ten. Independence would expect counts five point six two five, four point three seven five, three point three seven five, and two point six two five. At the observed table, where a equals eight, the conditional hypergeometric probability is about zero point zero two three six zero one. Across all seven feasible tables, only a equals three, eight, or nine is no more probable than the observation. Their probabilities sum to zero point zero three four nine seven. That exact p-value is below zero point zero five, so reject the fixed-margin independence null. The pilot supplies evidence of association, not causal proof. The sample odds ratio is eight times five over two times one, or twenty. With only sixteen observations, treat twenty as unstable and descriptive.

A reproducible analysis lists every feasible value from three through nine, its full table, its probability, and whether the two-sided rule includes it. Zero cells are valid counts, but they can make the sample odds ratio zero, infinite, or degenerate. Never hide that with an unreported correction. Validate nonnegative whole cells, at least one observation, positive row and column margins, total size at most five thousand, and alpha strictly between zero and one. Define both variables, category order, table orientation, analysis convention, and alpha before inspection. Relabeling a row or outcome afterward changes the claim. Route paired units, clusters, stratified designs, repeated looks, and tables larger than two-by-two to methods that preserve their actual data structure. Report the oriented table, margins, total, two-sided probability convention, alpha, exact p-value, sample odds ratio, design limits, and a guarded conclusion. Test a sparse two-by-two table, inspect every feasible result, and report the exact evidence free at Distri Scope dot com.