⚠️ Calculator not found

This calculator does not exist or may have been moved.

📊

Distribution

Chi-Square Calculator — Critical Value Table

Free chi-square (χ²) distribution calculator. Compute p-values from a χ² statistic and degrees of freedom, plus critical χ² values — for the test of independence, goodness-of-fit, and variance tests.

📊 χ² = 16.92, df = 9 → p = .0500

Got your number. Need a full analysis of your own data? ▶ Open MindStat

The chi-square (χ²) distribution arises as the sum of k independent squared standard-normal variables, which makes it the natural reference distribution for anything built from squared deviations: the chi-square test of independence for contingency tables, the goodness-of-fit test, and the confidence interval for a single population variance. Because it is a sum of squares, χ² is never negative and is right-skewed, especially at low degrees of freedom (df) — as df grows, it gradually approaches a normal shape. df is usually determined by the table's structure, e.g. (rows − 1) × (columns − 1) for a contingency table.

Chi-square goodness-of-fit

χ^{2} = \sum_{i = 1}^{k} \frac{{(O_{i} - E_{i})}^{2}}{E_{i}}

Critical χ² Values (upper tail)

df	χ².90 (α=.10)	χ².95 (α=.05)	χ².975 (α=.025)	χ².99 (α=.01)
1	2.706	3.841	5.024	6.635
2	4.605	5.991	7.378	9.210
3	6.251	7.815	9.348	11.345
4	7.779	9.488	11.143	13.277
5	9.236	11.070	12.833	15.086
6	10.645	12.592	14.449	16.812
7	12.017	14.067	16.013	18.475
8	13.362	15.507	17.535	20.090
9	14.684	16.919	19.023	21.666
10	15.987	18.307	20.483	23.209
15	22.307	24.996	27.488	30.578
20	28.412	31.410	34.170	37.566
25	34.382	37.652	40.646	44.314
30	40.256	43.773	46.979	50.892
40	51.805	55.758	59.342	63.691
50	63.167	67.505	71.420	76.154
60	74.397	79.082	83.298	88.379
80	96.578	101.879	106.629	112.329
100	118.498	124.342	129.561	135.807

Generated directly from src/core/distributions.js (chiInv) — never hand-typed.

Frequently Asked Questions

Why is chi-square always positive?

Because it is defined as a sum of squared terms — Σ(Oᵢ−Eᵢ)²/Eᵢ or a sum of squared standard-normal variables — and a square can never be negative. The distribution's support is [0, ∞), which is also why, unlike the normal or t distributions, chi-square tests are typically one-tailed (only large χ² values are 'unusual').

How do I find df for a contingency table?

For a chi-square test of independence on an r × c contingency table, df = (r − 1) × (c − 1). For example, a 2×2 table (two groups, two outcomes) has df = 1; a 3×4 table has df = 6. For a goodness-of-fit test against k categories, df = k − 1 (minus one more per estimated parameter, if any).

What does a large chi-square statistic mean?

A large χ² means the observed frequencies deviate substantially from what the null hypothesis (independence, or the specified distribution) predicts. Whether that deviation is 'significant' depends on comparing it to the critical value for the chosen α and df — this calculator's p-value (P(χ² ≥ x)) gives the exact probability of observing a deviation this large, or larger, purely by chance under the null.

Is chi-square the same as Cramér's V?

No — chi-square (χ²) is a test statistic that indicates whether an association exists and how likely it is by chance, but it grows mechanically with sample size and has no fixed upper bound, so it cannot be compared across studies. Cramér's V rescales χ² into a bounded [0, 1] effect-size measure, which is what should be reported and compared alongside the significance test.

← All Calculators 📄 Full Calculators Page

⚠️ Calculator not found

Chi-Square Calculator — Critical Value Table

Critical χ² Values (upper tail)

Frequently Asked Questions

Related Calculators