2x2 Chi Square Statistic Calculator

OBSERVED VS EXPECTED / 2X2 TABLE

See which cells are driving the chi-square statistic

Use this explainer when you need to audit a 2x2 table rather than stop at one test statistic. With the reviewed default counts a = 3, b = 2, c = 15, and d = 35, the calculator returns chi-square = 9.09090909 with 1 degree of freedom. The sections below show the row totals, column totals, expected counts, and cell-by-cell contributions behind that result.

Worked default example

Entered counts

Observed counts: a = 3, b = 2, c = 15, d = 35.

Row and column totals

Top row = 5, bottom row = 5, left column = 45, right column = 55, grand total = 1.

Expected counts under independence

Expected a = 22.5, b = 27.5, c = 22.5, d = 27.5. The smallest expected count is 22.5.

2x2 shortcut formula

The calculator's direct formula path is 1 x (3 x 35 - 2 x 15)^2 / [(5)(5)(45)(55)] = 9.09090909.

Equivalent Pearson sum

The same result also appears as (3 - 22.5)^2 / 22.5 + (2 - 27.5)^2 / 27.5 + (15 - 22.5)^2 / 22.5 + (35 - 27.5)^2 / 27.5 = 9.09090909.

What to enter and how to read the result

Enter four observed counts from one 2x2 contingency table. The main result card reports the uncorrected Pearson chi-square statistic, and the first supporting metric reports the degrees of freedom, which stays 1 for every 2x2 table. This page does not label your categories, calculate a p-value, or switch to a continuity-corrected or exact test.

What the calculator actually does

Nirmion evaluates the 2x2 shortcut formula n(ad-bc)^2 / [(a+b)(c+d)(a+c)(b+d)], which is algebraically equivalent to the usual sum of (observed - expected)^2 / expected across all four cells. In the reviewed default example, the largest contributions come from cells a and c at 2.5 each, while cells b and d contribute 2.045455 each, adding to chi-square = 9.09090909.

Where this model stops helping

Use raw counts, not percentages, rates, or rounded shares. The observations should be independent, and expected counts that are small, commonly below 5, weaken this uncorrected Pearson approximation. When that happens, consider Fisher's exact test, a continuity correction, or a fuller hypothesis-test workflow outside this calculator.

Reference and next checks

The linked OpenStax section explains the general chi-square test of independence, including expected-count construction and degrees of freedom. If you want a follow-up percentage check, a standard-score comparison, or a spread measure for numeric data, continue with the related calculators below.