Chi-Square Table: Critical Values for df 1 to 30

Q: What is the chi-square critical value at 5% with 10 degrees of freedom?

Row 10, column 0.05: 18.307. A test statistic above 18.307 on 10 df is significant at the 5% level.

Q: Why are there no negative chi-square values?

Chi-square statistics are sums of squared quantities (observed minus expected, squared, divided by expected). Squares cannot be negative, so the distribution starts at 0 and stretches right.

Q: Do I halve alpha for a chi-square test?

No. Chi-square significance is read from the right tail directly: the column headed by your alpha is the critical value. The variance confidence interval is the exception, using both tails deliberately.

Q: My df is above 30. What now?

The chi-square calculators compute the distribution for any df. For a table-based approximation beyond 30, the distribution approaches normal with mean df and variance 2df, but the calculators are the reliable route.

How to read this chi-square table

Columns are right-tail areas: the probability of a chi-square value at least as large as the cell, if the null hypothesis holds. Unlike the t-table, chi-square tests almost always work in the right tail and quote alpha directly: the 0.05 column is the 5% critical value, no halving, no doubling. Rows are degrees of freedom.

Example: a goodness-of-fit test on a die rolled 60 times gives χ² = 13.0 on 5 degrees of freedom. The 0.05 column at df 5 reads 11.070, and the 0.025 column reads 12.833. Your 13.0 clears both, so p < 0.025. It does not clear the 0.01 column (15.086). The exact p from the calculator is 0.0234, inside the table's bracket.

df	0.995	0.99	0.975	0.95	0.9	0.1	0.05	0.025	0.01	0.005
1	0.000	0.000	0.001	0.004	0.016	2.706	3.841	5.024	6.635	7.879
2	0.010	0.020	0.051	0.103	0.211	4.605	5.991	7.378	9.210	10.597
3	0.072	0.115	0.216	0.352	0.584	6.251	7.815	9.348	11.345	12.838
4	0.207	0.297	0.484	0.711	1.064	7.779	9.488	11.143	13.277	14.860
5	0.412	0.554	0.831	1.145	1.610	9.236	11.070	12.833	15.086	16.750
6	0.676	0.872	1.237	1.635	2.204	10.645	12.592	14.449	16.812	18.548
7	0.989	1.239	1.690	2.167	2.833	12.017	14.067	16.013	18.475	20.278
8	1.344	1.646	2.180	2.733	3.490	13.362	15.507	17.535	20.090	21.955
9	1.735	2.088	2.700	3.325	4.168	14.684	16.919	19.023	21.666	23.589
10	2.156	2.558	3.247	3.940	4.865	15.987	18.307	20.483	23.209	25.188
11	2.603	3.053	3.816	4.575	5.578	17.275	19.675	21.920	24.725	26.757
12	3.074	3.571	4.404	5.226	6.304	18.549	21.026	23.337	26.217	28.300
13	3.565	4.107	5.009	5.892	7.042	19.812	22.362	24.736	27.688	29.819
14	4.075	4.660	5.629	6.571	7.790	21.064	23.685	26.119	29.141	31.319
15	4.601	5.229	6.262	7.261	8.547	22.307	24.996	27.488	30.578	32.801
16	5.142	5.812	6.908	7.962	9.312	23.542	26.296	28.845	32.000	34.267
17	5.697	6.408	7.564	8.672	10.085	24.769	27.587	30.191	33.409	35.718
18	6.265	7.015	8.231	9.390	10.865	25.989	28.869	31.526	34.805	37.156
19	6.844	7.633	8.907	10.117	11.651	27.204	30.144	32.852	36.191	38.582
20	7.434	8.260	9.591	10.851	12.443	28.412	31.410	34.170	37.566	39.997
21	8.034	8.897	10.283	11.591	13.240	29.615	32.671	35.479	38.932	41.401
22	8.643	9.542	10.982	12.338	14.041	30.813	33.924	36.781	40.289	42.796
23	9.260	10.196	11.689	13.091	14.848	32.007	35.172	38.076	41.638	44.181
24	9.886	10.856	12.401	13.848	15.659	33.196	36.415	39.364	42.980	45.559
25	10.520	11.524	13.120	14.611	16.473	34.382	37.652	40.646	44.314	46.928
26	11.160	12.198	13.844	15.379	17.292	35.563	38.885	41.923	45.642	48.290
27	11.808	12.879	14.573	16.151	18.114	36.741	40.113	43.195	46.963	49.645
28	12.461	13.565	15.308	16.928	18.939	37.916	41.337	44.461	48.278	50.993
29	13.121	14.256	16.047	17.708	19.768	39.087	42.557	45.722	49.588	52.336
30	13.787	14.953	16.791	18.493	20.599	40.256	43.773	46.979	50.892	53.672

The left-hand columns earn their place too

Most classroom tables print only right-tail values, but the 0.95 and 0.99 columns (large areas to the right, small values of χ²) matter in two places: confidence intervals for a variance, which need both ends of the distribution, and the occasional test where a suspiciously small chi-square is the story (data that fit a model too well have their own history in science). The full spread printed here covers both.

Worked example

A 3-by-2 independence test returns χ² = 6.836 on 2 degrees of freedom. Row 2, 0.05 column: 5.991. The statistic clears it, so the association is significant at 5%. The 0.025 column (7.378) is not cleared, so p sits between 0.025 and 0.05; the calculator's exact value is 0.0328.

Degrees of freedom, by test

Goodness of fit: categories − 1 (minus any parameters estimated from the data). Independence in a contingency table: (rows − 1) × (columns − 1). Variance tests and intervals: n − 1. Getting df right matters more than reading the table precisely: the wrong row is a wrong answer no matter how carefully you read it.

Frequently asked questions

What is the chi-square critical value at 5% with 10 degrees of freedom?