What table formats are supported?

The tool accepts 2×2 and R×C tables, with rows on separate lines and cells separated by commas, spaces, or tabs.

When is Fisher's exact test recommended?

It is recommended for 2×2 tables when expected counts are small and the chi-square approximation may be unreliable.

What does the significance level α control?

Alpha sets the threshold for judging statistical significance, with a default of 0.05.

What does Cramér's V measure?

Cramér's V summarizes the strength of association between categorical variables.

What is the Chapman estimate used for?

The optional Chapman estimate provides a 2×2 mark-recapture population estimate using N̂ = (a+1)(b+1)/(c+1) − 1.

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Contingency Table Chi-Square Fisher Exact Cramér V Tutor

Step-by-step independence testing for 2×2 / R×C contingency tables: Pearson χ², Yates, G-test, Fisher exact, effect sizes, adjusted residuals, Cochran rule and optional Chapman estimate.

Details

What this tool helps you do

χ² = Σ(O−E)²/E; Yates for 2×2; G = 2ΣO·ln(O/E); p via regularized incomplete gamma; Fisher via lgamma hypergeometric with SciPy two-tailed convention; Cramér's V / phi / C; Haberman residuals; BH-FDR note; Chapman N̂ for 2×2 mark-recapture.

Execution

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2 examples

Test a balanced 2×2 table with textbook values

The table [[30,20],[20,30]] yields the exact textbook statistics χ²=4 (p=0.0455, significant at α=0.05), Yates χ²=3.24, G=4.0271, Cramér's V=0.2 and adjusted residuals of ±2 on every cell.

<section>χ² = 4 · df = 1 · p = 0.0455 (significant) · Yates 3.24 · G 4.0271 · V 0.2 · residuals ±2</section>

Small counts route to Fisher's exact test (tea tasting)

Fisher's lady-tasting-tea table [[3,1],[1,3]] has expected counts of 2 everywhere, so the Cochran rule flags it; the exact two-tailed probability 34/70 = 0.4857 shows no evidence against independence, in contrast to the unreliable χ² approximation.

<section>χ² = 2 · p = 0.1573 · Fisher two-tailed p = 0.4857 · E = 2 in all cells → exact test recommended</section>

Inputs

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4 options

ContentPaste or type the main input values.1

Contingency table (rows on lines, cells comma/space/tab-separated)textareaRequired

SettingsAdjust formats, ranges, numbers, and modes.2

Yates continuity correctionselectOptionalSignificance level αnumberOptional

TogglesEnable or disable optional behavior.1

Chapman capture-recapture estimate (2×2)checkboxOptionalEnabled when checkedFor 2×2 mark-recapture data: N̂ = (a+1)(b+1)/(c+1) − 1.

Result

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Samples

Contingency Table Chi-Square Fisher Exact Cramér V Tutor

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Key facts

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How it works

Use cases

Examples

1. Test a balanced 2×2 table

2. Use Fisher's exact test for small counts

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