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Chi-square and Fisher’s exact

Compare counts across categories, with the odds ratio and its interval.

Paste straight from a spreadsheet. Any table works; 2×2 also gets Fisher’s exact test and an odds ratio.

Paste a table of counts — at least two rows and two columns.

Formulaχ² = Σ (|O − E| − c)² ÷ E, c = ½ on 2×2; Fisher sums hypergeometric tables at least as extreme
ModelChi-square, uncorrected

Report the proportions as well. “12 of 40 against 31 of 44” tells a reader more than any p-value attached to it.

When to use this

Use this for counts in categories — how many colonies grew and did not, on each of two plates. Chi-square is the general test; Fisher’s exact is the one to use when any expected count is small, which in a typical biology table is often.

Worked example

Transformation efficiency with two different plasmids.

Table
2 × 2 counts
Test
Fisher’s exact

Result

Odds ratio with its confidence interval, and an exact p-value

The odds ratio is the effect size; an interval spanning 1 means the data are consistent with no difference.

What people get wrong

  • Using chi-square when expected counts are small. Below about five per cell the approximation breaks down and Fisher’s exact is the honest test.
  • Putting percentages into the table. These tests need raw counts; percentages discard the sample size the whole inference rests on.
  • Testing a table whose cells are not independent. Repeated measures on the same animals need a different approach entirely.

Questions

+Chi-square or Fisher’s exact?

Fisher’s exact when any expected count is below about five, and it is always defensible for a 2 × 2 table. Chi-square scales better to larger tables.

+What effect size should I report?

The odds ratio or the risk difference, with an interval. A p-value alone says nothing about how large the difference is.

+Do I need Yates’ correction?

It is conservative and largely superseded by simply using Fisher’s exact when the counts are small.

Related tools

  • t-test calculator — Compare two groups and get the effect size and interval, not just a p-value.

Science last reviewed .