Chi-Square Calculator

This free chi square calculator runs a chi-square test of independence on any contingency table up to 12 × 12, a chi-square goodness-of-fit test against equal, custom-count or ratio expectations (like 9:3:3:1), and finds the p-value from a chi-square statistic or the critical value for any df and α. You get χ², degrees of freedom, the p-value, a reject or fail-to-reject decision, a table of expected counts with each cell’s contribution and adjusted residual, Cramér’s V, a Yates correction and Fisher exact p-value for 2 × 2 tables, a shaded chi-square curve, full step-by-step working and the exact TI-84 keystrokes for χ²-Test and χ²GOF-Test.

Contents

How to Use the Chi-Square Calculator

1. Pick the test. Test of independence for a two-way table, Goodness of fit for one list of counts, or P-value / critical value if you already have χ² and df.
2. Enter counts. For a table, type one row per line with columns separated by spaces, commas or tabs. You can paste a block straight from Excel or Google Sheets. For goodness of fit, list the observed counts and choose equal expected values, expected counts, or proportions, percentages or a ratio.
3. Choose α. Type any significance level or tap 0.10, 0.05, 0.025, 0.01 or 0.001.
4. Read the answer. The results update as you type: p-value, χ², df, critical value, the decision, an expected-count table with a heat map of contributions, and the TI-84 steps. Copy share link saves your exact inputs in the URL.

Chi-Square Formula and Degrees of Freedom

Every chi-square test compares observed counts O with the counts E you would expect if the null hypothesis were true:

χ² = Σ (O − E)² / E

Test	Expected count E	Degrees of freedom	H₀
Goodness of fit	Ei = n × pi	k − 1	The data follow the stated distribution
Independence / homogeneity	Eij = (row total × column total) / n	(r − 1)(c − 1)	The two variables are independent

The p-value is the right-tail area P(χ²_df ≥ your χ²). Chi-square tests are always right-tailed, because any mismatch between O and E makes χ² bigger.

Chi-Square Goodness-of-Fit Test (Worked Example)

Is a die fair? It is rolled 60 times and gives 8, 9, 19, 5, 8 and 11 for faces 1–6. If it is fair, each face is expected 60 / 6 = 10 times.

Face	1	2	3	4	5	6
Observed O	8	9	19	5	8	11
Expected E	10	10	10	10	10	10
(O − E)²/E	0.4	0.1	8.1	2.5	0.4	0.1

χ² = 0.4 + 0.1 + 8.1 + 2.5 + 0.4 + 0.1 = 11.6 with df = 6 − 1 = 5. The p-value is P(χ²₅ ≥ 11.6) = 0.0407. Because 0.0407 ≤ 0.05 (and 11.6 > the critical value 11.070), we reject H₀: there is evidence the die is not fair. Face 3 contributes most of the statistic.

Ratio example (Mendel’s peas): observed 315, 108, 101, 32 against a 9:3:3:1 ratio. Expected counts are 312.75, 104.25, 104.25 and 34.75, so χ² = 0.470, df = 3, p = 0.9254. We fail to reject H₀; the data fit the 9:3:3:1 ratio well. Tap Mendel 9:3:3:1 example in the calculator to see it.

Chi-Square Test of Independence (Worked Example)

A survey asks 395 people (two groups) which of four options they prefer:

	A	B	C	D	Total
Group 1	60	54	46	41	201
Group 2	40	44	53	57	194
Total	100	98	99	98	395

1. Expected counts: E = row total × column total / n. For Group 1, A: 201 × 100 / 395 = 50.886.
2. Contribution: (60 − 50.886)² / 50.886 = 1.632. Repeat for all 8 cells.
3. Add them: χ² = 8.006, df = (2 − 1)(4 − 1) = 3.
4. P-value: P(χ²₃ ≥ 8.006) = 0.0459, below α = 0.05, so reject H₀: preference is associated with group.
5. Strength: Cramér’s V = √(8.006 / (395 × 1)) = 0.142, a small association.

This is the default example in the calculator, so you can check each cell’s expected count and adjusted residual there. The same test is called the chi-square test of homogeneity when the rows are separate samples; the arithmetic is identical.

How to Find the P-Value from a Chi-Square Statistic

Already have the statistic? Open the P-value / critical value tab, enter χ² and df, and read the right-tail p-value, the left area and a row of critical values. By hand or on a TI-84, p-value = χ²cdf(χ², 1E99, df). For example, χ² = 7.815 with df = 3 gives p ≈ 0.0500, and χ² = 3.841 with df = 1 gives p ≈ 0.0500.

Chi-Square Critical Value Table

Reject H₀ when your χ² is greater than or equal to the critical value for your df and α (right-tail area).

df	α = 0.10	α = 0.05	α = 0.025	α = 0.01	α = 0.005	α = 0.001
1	2.706	3.841	5.024	6.635	7.879	10.828
2	4.605	5.991	7.378	9.210	10.597	13.816
3	6.251	7.815	9.348	11.345	12.838	16.266
4	7.779	9.488	11.143	13.277	14.860	18.467
5	9.236	11.070	12.833	15.086	16.750	20.515
6	10.645	12.592	14.449	16.812	18.548	22.458
7	12.017	14.067	16.013	18.475	20.278	24.322
8	13.362	15.507	17.535	20.090	21.955	26.124
9	14.684	16.919	19.023	21.666	23.589	27.877
10	15.987	18.307	20.483	23.209	25.188	29.588
12	18.549	21.026	23.337	26.217	28.300	32.909
15	22.307	24.996	27.488	30.578	32.801	37.697
20	28.412	31.410	34.170	37.566	39.997	45.315
25	34.382	37.652	40.646	44.314	46.928	52.620
30	40.256	43.773	46.979	50.892	53.672	59.703

How to Do a Chi-Square Test on a TI-84

χ² test of independence (χ²-Test)

1. Press 2nd x⁻¹ (MATRIX), go to EDIT, choose 1:[A] and set its size to rows × columns.
2. Type the observed counts row by row.
3. Press STAT, go to TESTS, choose C:χ²-Test. Observed: [A], Expected: [B].
4. Choose Calculate. The calculator shows χ², p and df, and stores the expected counts in [B].

χ² goodness of fit (χ²GOF-Test)

1. Press STAT 1:Edit. Put the observed counts in L1 and the expected counts (not proportions) in L2.
2. Press STAT, go to TESTS, choose D:χ²GOF-Test. Observed: L1, Expected: L2, df: k − 1.
3. Choose Calculate to see χ², p, df and the CNTRB list of contributions. χ²GOF-Test needs a TI-84 Plus OS 2.30 or later; the TI-83 does not have it.

For a p-value only, use 2nd VARS χ²cdf(lower, 1E99, df). Practice on our TI-84 Plus CE calculator online.

Assumptions: Expected Counts, Yates and Fisher

• Counts, not percentages. Enter raw frequencies. Percentages or means give wrong answers.
• Independent observations. Each person or item belongs in exactly one cell.
• Expected counts of at least 5. A common rule is that all expected counts are ≥ 1 and no more than 20% are below 5. The calculator flags low cells for you.
• Yates continuity correction subtracts 0.5 from each |O − E| in a 2 × 2 table. It is more conservative. The TI-84 does not apply it, so leave it off to match your calculator.
• Fisher’s exact test is shown automatically for 2 × 2 tables of whole numbers. Use it when expected counts are small.

Effect Size: Cramér’s V

A small p-value says an association exists, not how strong it is. Cramér’s V = √(χ² / (n · (min(r, c) − 1))) runs from 0 (no association) to 1 (perfect association). For a 2 × 2 table it equals the phi coefficient. As a rough guide with df* = 1: 0.1 is small, 0.3 medium and 0.5 large. The calculator labels the size for your table.

Why This Chi-Square Calculator Is Different

• All three chi-square tools in one place: independence, goodness of fit, and p-value or critical value from χ².
• Paste a whole table from a spreadsheet, up to 12 × 12, with optional row and column labels.
• Goodness of fit accepts expected counts, proportions, percentages or ratios like 9:3:3:1, and rescales them to your total.
• Expected-count table with a heat map of each cell’s contribution plus adjusted residuals, so you can see which cells drive the result.
• Cramér’s V, Yates correction and Fisher exact p-value for 2 × 2 tables, with automatic low-expected-count warnings.
• Shaded chi-square curve, step-by-step working, a plain-English conclusion and TI-84 keystrokes with the exact screen output.
• Checked against SciPy on more than 11,000 values. Share links, dark mode and a phone-friendly layout. See all tools on the math and statistics calculators page, or run z and t tests with the hypothesis test calculator.

Chi-Square Calculator FAQ

Ti84CalcPro is not affiliated with or endorsed by Texas Instruments. TI-84 Plus CE is a trademark of Texas Instruments.