Chi-Square Test Calculator

Run a chi-square test of independence on counts, or of goodness of fit, with Yates' correction, Fisher's exact test for 2 × 2 tables and Cramér's V.

Result

Deterministic calculation

χ² statistic

χ²(2) = 10.937

p-value: 0.00422
Effect size, Cramér's V: 0.312 (one of the two variables has two levels)
Total count, table size: 112, 2 × 3

Expected counts under independence
	Column 1	Column 2	Column 3
Row 1	17.41	23.21	24.38
Row 2	12.59	16.79	17.63

Note

The p-value is the probability, if the null hypothesis and every assumption of the test were true, of a result at least as extreme as the one observed. It is not the probability that the hypothesis is true, and it says nothing about how large or important an effect is. Judge the effect size and its interval, the design and the assumptions along with it, and do not treat a p-value on one side of 0.05 as proof of an effect or of its absence.

Note

A test of independence says whether the two variables are associated, not how strongly, which is what Cramér's V (0 to 1) measures, nor why. Observations must be independent: repeated measurements on the same subjects violate that.

Significant figuresScientific notation

Calculation

expected count = row total × column total / grand total
χ² = Σ (observed − expected)² / expected, with (rows − 1)(columns − 1) = 2 degrees of freedom
Cramér's V = √(χ² / (N × min(rows − 1, columns − 1)))

Formula

\chi^2 = \sum \dfrac{(O - E)^2}{E}

E = \dfrac{\text{row total}\times\text{column total}}{\text{grand total}}

V = \sqrt{\dfrac{\chi^2}{N\,\min(r - 1,\ c - 1)}}

$\text{df}$: (rows − 1)(columns − 1) for a table; categories − 1 for goodness of fit
$V$: effect size from 0 (no association) to 1 (complete)

How it works

A chi-square test compares observed counts with the counts expected if there were no association (or if the data followed the expected proportions). Large departures give a large χ², and so a small p-value. It applies to counts of independent observations, not to percentages or measurements.

The chi-square approximation is reliable only when expected counts are not small, commonly no more than a fifth of them below 5. For a 2 × 2 table with small counts, Fisher's exact test is preferred; Yates' continuity correction is an older adjustment that makes the test conservative. Cramér's V, which goes from 0 to 1, says how strong the association is, which the p-value does not.

Worked example

A 2 × 3 table: row 1 has 20, 15 and 30; row 2 has 10, 25 and 12.

Expected counts: row 1 is 17.41, 23.21 and 24.38; row 2 is 12.59, 16.79 and 17.63.
χ² = 10.937 with 2 degrees of freedom; p = 0.0042; V = 0.312.

The pattern across columns differs between the rows, with a moderate association (V = 0.31); all expected counts exceed 5.

These are the values the calculator opens with, so you can check its output against this example.

Assumptions

Independent observations, each in exactly one cell.
Expected counts not too small for the approximation.
Counts, not proportions.

Common mistakes

Entering percentages instead of counts.
Using the chi-square approximation on sparse tables.
Using it for paired or repeated data, where observations are not independent.

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Chi-Square Test Calculator

Result

Calculation

Formula

How it works

Worked example

Assumptions

Common mistakes

Related tools

Result

Calculation