Reference · Statistics

Chi-Square Table

Critical χ² values for every degrees of freedom and upper-tail area, with your row highlighted.

Degrees of freedom

For a goodness-of-fit test df = categories − 1; for a contingency table df = (rows − 1) × (columns − 1).

Critical χ² at α = 0.05, df = 1

3.841

Critical χ² at α = 0.01, df = 1

6.635

Chi-square critical values by degrees of freedom and upper-tail area
df	0.995	0.990	0.975	0.950	0.900	0.100	0.050	0.025	0.010	0.005
1	3.93e-5	1.57e-4	9.82e-4	0.0039	0.0158	2.706	3.841	5.024	6.635	7.879
2	0.0100	0.0201	0.0506	0.1026	0.2107	4.605	5.991	7.378	9.210	10.597
3	0.0717	0.1148	0.2158	0.3518	0.5844	6.251	7.815	9.348	11.345	12.838
4	0.2070	0.2971	0.4844	0.7107	1.064	7.779	9.488	11.143	13.277	14.860
5	0.4117	0.5543	0.8312	1.145	1.610	9.236	11.070	12.833	15.086	16.750
6	0.6757	0.8721	1.237	1.635	2.204	10.645	12.592	14.449	16.812	18.548
7	0.9893	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
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	100.43	104.21
80	51.172	53.540	57.153	60.391	64.278	96.578	101.88	106.63	112.33	116.32
90	59.196	61.754	65.647	69.126	73.291	107.57	113.15	118.14	124.12	128.30
100	67.328	70.065	74.222	77.929	82.358	118.50	124.34	129.56	135.81	140.17

Every column heading is the area in the upper tail — the probability of a χ² larger than that value. The first five columns (0.995 to 0.90) are the lower-tail critical values used for variance confidence intervals, not for goodness-of-fit tests.

A chi-square table gives the cut-off a χ² statistic must exceed to be significant. Find your degrees of freedom down the side and the tail area across the top: at df = 3 and α = 0.05 the critical value is 7.815, so a χ² above 7.815 rejects the null hypothesis.

What the chi-square distribution describes

Chi-square is the distribution of a sum of squared standard normal values. Because everything in it is squared it can never be negative, and unlike the normal or t curves it is not symmetric — it is skewed to the right, heavily so at small degrees of freedom, and slowly straightens out as df grows. That asymmetry is why its table has separate lower- and upper-tail columns rather than a single ± value.

Two very common tests use it. A goodness-of-fit test asks whether observed counts match an expected distribution, with df = categories − 1. A test of independence asks whether two categorical variables are related in a contingency table, with df = (rows − 1) × (columns − 1). Both compare observed against expected counts and both are one-tailed: only a large χ² is evidence against the null.

Why the first columns look backwards

The columns headed 0.995 through 0.90 are lower-tail cut-offs, and they hold small numbers. They are not used for goodness-of-fit tests — they exist for confidence intervals on a variance, which need a value from each end of the distribution. For an ordinary significance test you want the right-hand columns.

χ² = Σ (O − E)² ÷ E

O the observed count in a category, E the expected count, summed over every category

1
Work out the degrees of freedom. Testing whether a die is fair compares six categories, so df = 6 − 1 = 5.
2
Choose the significance level. At α = 0.05 you are asking for the value with 5% of the distribution beyond it.
3
Read across the row to that column. Row df = 5 and column 0.05 give a critical value of 11.070.
4
Compute your test statistic. Sum (O − E)² ÷ E across the six faces; suppose that comes to 12.40.
5
Compare and conclude. Because 12.40 is larger than 11.070, the counts differ from a fair die by more than chance would explain at the 5% level.

Critical χ² values at the common levels

Upper-tail cut-offs — reject the null hypothesis when your statistic exceeds the value shown.

df	α = 0.05	α = 0.01
1	3.841	6.635
2	5.991	9.210
3	7.815	11.345
4	9.488	13.277
5	11.070	15.086
6	12.592	16.812
7	14.067	18.475
8	15.507	20.090
9	16.919	21.666
10	18.307	23.209

Conditions the test needs

The chi-square approximation assumes reasonably large expected counts. The usual rule is that every expected count should be at least 5, or at minimum that no more than a fifth of them fall below 5 and none is zero. With sparse tables the approximation drifts and an exact test — Fisher’s exact test for a 2×2 table — is the safer choice.

Two other habits matter. The test works on raw counts, never on percentages or averages, so converting to proportions first invalidates it. And a significant result says only that observed and expected differ by more than chance; it does not say which cell caused it or how large the effect is. Inspect the individual (O − E)² ÷ E contributions to see where the difference lives, and report an effect size such as Cramér’s V alongside the p-value.

How do I calculate degrees of freedom for a chi-square test?

For a goodness-of-fit test df = categories − 1. For a test of independence on a contingency table df = (rows − 1) × (columns − 1), so a 3×4 table has df = 2 × 3 = 6.

Why is the chi-square test always one-tailed?

The statistic sums squared differences, so it only grows as observed counts drift from expected ones. Large values mean poor fit and small values mean close fit, which is why only the upper tail carries evidence against the null.

What do the 0.995 and 0.99 columns mean?

They are lower-tail cut-offs, used when you need a value from the left end of the distribution — chiefly for a confidence interval on a variance or standard deviation. They are not the critical values for a goodness-of-fit or independence test.

What is the critical value for a 2×2 contingency table?

A 2×2 table has df = (2 − 1) × (2 − 1) = 1, so the critical value at α = 0.05 is 3.841 and at α = 0.01 it is 6.635. The same 3.841 appears often because it is simply 1.96 squared.

What if some expected counts are below 5?

The chi-square approximation becomes unreliable. Combine sparse categories where it makes sense, collect more data, or switch to an exact test — Fisher’s exact test is the standard alternative for a 2×2 table.

Can the chi-square statistic be negative?

No. Every term in the sum is a squared difference divided by a positive expected count, so the statistic starts at zero and only increases. A negative result always means an arithmetic slip.

Chi-Square Table

What the chi-square distribution describes

Why the first columns look backwards

Critical χ² values at the common levels

Conditions the test needs

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