Fisher's Exact Test vs Chi-Square: When Is the Exact Test Required?
Short answer. Use Fisher's exact test when any expected count in the table is below 5. The chi-square approximation needs larger expected counts to stay accurate. With 7/1 against 1/7 the minimum expected count is 4.0, chi-square gives p=0.0027 and Fisher gives p=0.0101, so the two tests disagree at alpha 0.01.
The expected-count rule
Every cell has an expected count: row total times column total divided by the grand total. The usual rule of thumb, from Cochran's 1954 note on the chi-square test, is that no expected count should fall below 5 in a 2x2 table. When one does, the chi-square p value can be too small.
Fisher's exact test does not use that approximation. It fixes the row and column totals and sums the probability of every table at least as extreme as the observed one.
Worked example
Two small groups, 8 people each, with a yes/no outcome. Group A has 7 yes and 1 no. Group B has 1 yes and 7 no. Every expected count is 4.0, below the threshold.
Pearson chi-square, 1 df: chi2 = 9.0000, p = 0.0027
Fisher exact, two-sided: p = 0.0101Both tests reject at alpha 0.05. At alpha 0.01 they disagree: the approximate p value crosses the line and the exact one does not. The exact p value is the honest one here, because the table is too small for the approximation.
Why the exact test is conservative
The exact test sums probabilities of discrete tables, so the attainable p values are a finite set. It cannot return every value between 0 and 1. This makes it conservative: it holds the false-positive rate at or below the chosen alpha rather than exactly at it. A result that is only just significant under Fisher is weaker evidence than the same p value from a large-sample test.
Larger than 2x2
For a 2x3 or 3x3 table with sparse cells there is no single Fisher p value in the original sense. Use the Fisher-Freeman-Halton extension, which orders r by c tables by probability. The Fisher-Freeman-Halton calculator runs that test and shows the fixed margins, the expected counts and every tail table it summed.
For a 2x2 table the chi-square test calculator reports Pearson's statistic and its p value, so you can see how far the approximation moves.
Do not choose the test by the p value
Run the test the design calls for. Choosing between chi-square and Fisher after seeing which one is significant is a form of selection that inflates the error rate. State the rule before you look at the data: expected counts at or above 5 in every cell means chi-square is acceptable, and any cell below 5 means the exact test.
Related reading
Paired yes/no data needs a different test again, covered in McNemar test vs chi-square.