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Chi-Square Test Calculator

Q: Should I use Yates' correction?

Only for 2×2 tables and mainly with small samples; it makes the test conservative. Report which version you used, since R applies it by default and many other tools do not.

Q: How do I enter the table?

Row by row: a 2×2 table with rows (10, 20) and (30, 40) is observed = 10, 20, 30, 40 with rows = 2 and columns = 2.

Q: What does Cramér's V mean?

A 0–1 measure of association strength independent of sample size; roughly 0.1 small, 0.3 medium, 0.5 large for df* = 1 (Cohen), smaller thresholds for larger tables.

Computes Pearson's chi-square statistic, degrees of freedom, p-value, critical value and expected counts for a contingency table (test of independence, with Cramér's V) or for observed versus expected category counts (goodness-of-fit test).

Observed countsObserved frequencies. For a contingency table list the cells row by row (row-major) and give rows and columns; for goodness of fit list one count per category. Rows (optional)Number of rows of the contingency table (test of independence). Omit for goodness of fit. Columns (optional)Number of columns of the contingency table. Expected counts or proportions (optional)Goodness of fit only: expected count (or proportion) per category, same order as observed; rescaled to the observed total. Omit for equal expected counts. Yates' continuity correction (optional)Apply the continuity correction (|O − E| − 0.5)² to a 2×2 table, as R's chisq.test does by default. Ignored for other tables. Significance level (α) (optional)Significance level as a probability (0.05 = 5 %), not a percentage. Confidence intervals and critical values use level 1 − alpha.

When to use

You have counts in categories and want to know whether two categorical variables are associated, or whether observed frequencies match a hypothesised distribution.

Do not use when: Expected counts are below 5 in many cells (use Fisher's exact test), the data are means rather than counts (use t-test), or you compare exactly two proportions and want a confidence interval for their difference (use proportion-z-test).

Formula

χ² = Σ (O − E)² / E; independence: E_ij = (row total_i × column total_j) / N, df = (rows − 1)(columns − 1), Cramér's V = √(χ² / (N · min(rows − 1, columns − 1))); goodness of fit: E_i = N × p_i, df = k − 1; p = P(χ²_df ≥ χ²)

Pearson's chi-square with the asymptotic chi-square distribution (regularized incomplete gamma function). Rule of thumb: all expected counts ≥ 5 (at least 80 % of cells ≥ 5 and none below 1); a note is added otherwise.

Inputs

Parameter	Type	Required	Description
`observed`	number_list	yes	Observed frequencies. For a contingency table list the cells row by row (row-major) and give rows and columns; for goodness of fit list one count per category.
`rows`	integer	no	Number of rows of the contingency table (test of independence). Omit for goodness of fit. Range: ≥ 2, ≤ 60
`columns`	integer	no	Number of columns of the contingency table. Range: ≥ 2, ≤ 60
`expected`	number_list	no	Goodness of fit only: expected count (or proportion) per category, same order as observed; rescaled to the observed total. Omit for equal expected counts.
`yates_correction`	boolean	default false	Apply the continuity correction (\|O − E\| − 0.5)² to a 2×2 table, as R's chisq.test does by default. Ignored for other tables.
`alpha`	number	default 0.05	Significance level as a probability (0.05 = 5 %), not a percentage. Confidence intervals and critical values use level 1 − alpha. Range: > 0

Outputs

Output	Type	Description
`test`	string	Test of independence (with table size) or goodness of fit.
`chi_square`	number	Σ (O − E)² / E over all cells.
`degrees_of_freedom`	integer	(rows − 1)(columns − 1) for independence; categories − 1 for goodness of fit.
`p_value`	number	Upper-tail probability P(χ²_df ≥ chi_square).
`critical_value`	number	Chi-square quantile at 1 − alpha; H0 is rejected when chi_square exceeds it.
`cramers_v`	number	Effect size √(χ² / (N · min(rows − 1, columns − 1))), 0 to 1 (independence test only; equals phi for 2×2).
`expected_counts`	number_list	Expected frequency of every cell under H0, in the same order as observed.
`min_expected_count`	number	The chi-square approximation is questionable when this is below 5.
`significant`	boolean	true when p_value < alpha.
`decision`	string	Plain-language conclusion.

Example

Independence: 3 occupation classes × 4 neighbourhoods (Wikipedia example, N = 650): {"observed":[90,60,104,95,30,50,51,20,30,40,45,35],"rows":3,"columns":4} → {"chi_square":24.5712,"degrees_of_freedom":6,"p_value":0.00041,"critical_value":12.5916,"cramers_v":0.1375,"min_expected_count":34.62,"significant":true}

Goodness of fit: die rolled 60 times, 5/8/9/8/10/20: {"observed":[5,8,9,8,10,20]} → {"chi_square":13.4,"degrees_of_freedom":5,"p_value":0.019905,"critical_value":11.0705,"expected_counts":[10,10,10,10,10,10],"significant":true}

GET https://tttkmbb.com/api/v1/calculate/chi-square-test?observed=90%2C60%2C104%2C95%2C30%2C50%2C51%2C20%2C30%2C40%2C45%2C35&rows=3&columns=4

Machine access

API: GET https://tttkmbb.com/api/v1/calculate/chi-square-test (query parameters) or POST with a JSON body {"inputs": {...}}
Schema: https://tttkmbb.com/api/v1/calculators/chi-square-test · Markdown: https://tttkmbb.com/statistics/chi-square-test.md · JSON definition: https://tttkmbb.com/statistics/chi-square-test.json
MCP: server https://tttkmbb.com/mcp, tool run_calculator with calculator_id="chi-square-test"
OpenAPI operationId: calculate_chi_square_test
Freshness: static. Authentication: none. Rate limit: fair use (see rate limits).

Sources

NIST/SEMATECH e-Handbook of Statistical Methods, 1.3.5.15 Chi-Square Goodness-of-Fit Test (government)
Wikipedia – Pearson's chi-squared test (reference)
Wikipedia – Cramér's V (reference)

FAQ

Should I use Yates' correction?

Only for 2×2 tables and mainly with small samples; it makes the test conservative. Report which version you used, since R applies it by default and many other tools do not.

How do I enter the table?

Row by row: a 2×2 table with rows (10, 20) and (30, 40) is observed = 10, 20, 30, 40 with rows = 2 and columns = 2.

What does Cramér's V mean?

A 0–1 measure of association strength independent of sample size; roughly 0.1 small, 0.3 medium, 0.5 large for df* = 1 (Cohen), smaller thresholds for larger tables.

Related calculators

P-Value Calculator — p-value from a chi-square statistic and df.
Two-Proportion Z-Test Calculator — Compare two proportions with a confidence interval for the difference.
Odds Ratio Calculator — Effect measures for a 2×2 table.