Fischer Exact Test

The row and column totals remain fixed while the test evaluates how likely a particular arrangement of counts would be under the null hypothesis. This constraint defines the set of tables being compared and allows the probability calculation to use the hypergeometric distribution. Consequently, the assessment focuses on the allocation of observations between categories rather than changing overall group or category totals.

The hypergeometric distribution supplies the probability for the observed 2 × 2 table under the assumption that the categorical variables are unrelated. The calculation then includes the observed arrangement and tables considered more extreme. Because these probabilities are computed directly rather than estimated through a large-sample approximation, the result remains useful for small datasets and low expected cell counts.

The main distinction is how each method obtains its evidence. Fischer Exact Test calculates probabilities directly with fixed marginal totals, whereas the chi-square test relies on a large-sample approximation. When a table contains few observations or low expected cell counts, the exact approach avoids depending on an approximation that may be less suitable for the data structure.

The reported probability is not limited to the exact table observed. It also accounts for arrangements judged more inconsistent with the null hypothesis of no association, while preserving the marginal totals. This broader comparison determines the evidence considered by the test and helps researchers evaluate whether the observed relationship between classifications is statistically supported.

First, organize observations for two categorical variables in a 2 × 2 contingency table. Next, identify the null hypothesis of no association and retain the table’s marginal totals. The exact calculation uses the hypergeometric distribution to evaluate the observed arrangement and more extreme arrangements. Finally, interpret the resulting probability in relation to the proposed association.

The method is useful when researchers compare classifications between groups and the resulting contingency table is small or contains low expected cell counts. In biology, medicine, and social science, it can assess whether two categorical classifications are associated without requiring a large sample. Its outcome helps determine whether the observed difference between groups has statistical support.

It evaluates whether the observed allocation of cases across the categories is compatible with a null hypothesis of no association. A probability based on the observed and more extreme tables provides the statistical evidence for that assessment. The result supports interpretation of relationships between classifications, but it does not replace clear identification of the variables and their category structure.