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Chi-square table

Critical values of χ². Column headings are the area to the right (upper-tail probability).

Look up any df

Degrees of freedom

Area to the right

Or: your χ² statistic

Download the printable chi-square table (PDF, 1 page) →

Critical values of χ²

df \ area right	0.995	0.99	0.975	0.95	0.9	0.1	0.05	0.025	0.01	0.005
1	3.93e-5	1.57e-4	9.82e-4	0.0039	0.0158	2.7055	3.8415	5.0239	6.6349	7.8794
2	0.0100	0.0201	0.0506	0.1026	0.2107	4.6052	5.9915	7.3778	9.2103	10.597
3	0.0717	0.1148	0.2158	0.3518	0.5844	6.2514	7.8147	9.3484	11.345	12.838
4	0.2070	0.2971	0.4844	0.7107	1.0636	7.7794	9.4877	11.143	13.277	14.860
5	0.4117	0.5543	0.8312	1.1455	1.6103	9.2364	11.070	12.833	15.086	16.750
6	0.6757	0.8721	1.2373	1.6354	2.2041	10.645	12.592	14.449	16.812	18.548
7	0.9893	1.2390	1.6899	2.1673	2.8331	12.017	14.067	16.013	18.475	20.278
8	1.3444	1.6465	2.1797	2.7326	3.4895	13.362	15.507	17.535	20.090	21.955
9	1.7349	2.0879	2.7004	3.3251	4.1682	14.684	16.919	19.023	21.666	23.589
10	2.1559	2.5582	3.2470	3.9403	4.8652	15.987	18.307	20.483	23.209	25.188
11	2.6032	3.0535	3.8157	4.5748	5.5778	17.275	19.675	21.920	24.725	26.757
12	3.0738	3.5706	4.4038	5.2260	6.3038	18.549	21.026	23.337	26.217	28.300
13	3.5650	4.1069	5.0088	5.8919	7.0415	19.812	22.362	24.736	27.688	29.819
14	4.0747	4.6604	5.6287	6.5706	7.7895	21.064	23.685	26.119	29.141	31.319
15	4.6009	5.2293	6.2621	7.2609	8.5468	22.307	24.996	27.488	30.578	32.801
16	5.1422	5.8122	6.9077	7.9616	9.3122	23.542	26.296	28.845	32.000	34.267
17	5.6972	6.4078	7.5642	8.6718	10.085	24.769	27.587	30.191	33.409	35.718
18	6.2648	7.0149	8.2307	9.3905	10.865	25.989	28.869	31.526	34.805	37.156
19	6.8440	7.6327	8.9065	10.117	11.651	27.204	30.144	32.852	36.191	38.582
20	7.4338	8.2604	9.5908	10.851	12.443	28.412	31.410	34.170	37.566	39.997
21	8.0337	8.8972	10.283	11.591	13.240	29.615	32.671	35.479	38.932	41.401
22	8.6427	9.5425	10.982	12.338	14.041	30.813	33.924	36.781	40.289	42.796
23	9.2604	10.196	11.689	13.091	14.848	32.007	35.172	38.076	41.638	44.181
24	9.8862	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
40	20.707	22.164	24.433	26.509	29.051	51.805	55.758	59.342	63.691	66.766
50	27.991	29.707	32.357	34.764	37.689	63.167	67.505	71.420	76.154	79.490
60	35.534	37.485	40.482	43.188	46.459	74.397	79.082	83.298	88.379	91.952
70	43.275	45.442	48.758	51.739	55.329	85.527	90.531	95.023	100.425	104.215
80	51.172	53.540	57.153	60.391	64.278	96.578	101.879	106.629	112.329	116.321
90	59.196	61.754	65.647	69.126	73.291	107.565	113.145	118.136	124.116	128.299
100	67.328	70.065	74.222	77.929	82.358	118.498	124.342	129.561	135.807	140.169

Computed from the chi-square distribution (regularized incomplete gamma function) and checked against published tables in our test suite.

Use it with: MTBF confidence bound · Chi-square test guide · Confidence intervals

FAQ

How do I read a chi-square table?

Columns are the area to the right of the value. For a goodness-of-fit or independence test at α = 0.05, use the 0.05 column: reject if your statistic is larger. For a confidence interval on a variance you need two columns, α/2 and 1 − α/2.

What degrees of freedom do I use?

Goodness of fit: categories − 1 − parameters estimated. Contingency table: (rows − 1) × (columns − 1). Variance test or interval: n − 1. MTBF confidence bound: 2r + 2 for a time-terminated test.

How do I get a confidence interval for a standard deviation?

(n − 1)s² / χ²(α/2) ≤ σ² ≤ (n − 1)s² / χ²(1 − α/2), using the upper-tail α/2 and 1 − α/2 values for df = n − 1. Take square roots for σ.

Why are the values so small on the left?

The left columns are lower-tail critical values: almost all of the distribution lies above them. They are used for the lower end of variance intervals and for left-tailed variance tests.

Chi-square shows up in tests, intervals and reliability

This shows up on the exams as a calculation question. Practice it with new numbers every time, and every wrong answer explained.

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