Statistics Reference

Chi-Square Table (χ² Distribution Critical Values)

The chi-square table lists critical values of the χ² distribution: the cutoffs a chi-square statistic must exceed to be significant at a given level. Rows are degrees of freedom (df), columns are upper-tail areas. Because the distribution is not symmetric, the table prints both the small lower critical values (columns 0.995 to 0.90) and the familiar upper critical values (columns 0.10 to 0.005).

Tables by Significance Level

Need one level at every degree of freedom from 1 to 100? Each of the four standard tail areas has a dedicated page with both the upper and lower critical values:

α = 0.10 (90% confidence)

Screening-level goodness-of-fit and independence tests at α = 0.10

α = 0.05 (95% confidence)

Goodness-of-fit and independence tests at the conventional 5% level

α = 0.025 (97.5% confidence)

95% confidence intervals for a variance or standard deviation (0.025 per tail)

α = 0.01 (99% confidence)

Confirmatory chi-square tests at the strict 1% level

How to Read the Chi-Square Table

You need two things: the degrees of freedom for your procedure and the tail area (for a standard test, the significance level α). The critical value sits where the df row meets the column.

Example: goodness-of-fit test over four categories

Degrees of freedom: df = k − 1 = 4 − 1 = 3, so use the row labeled 3.
Testing at α = 0.05 puts all of the rejection area in the upper tail, so read the 0.05 column.
The cell gives 7.815. Reject the null hypothesis when your χ² statistic exceeds 7.815.

The same lookup works for independence tests: a 3×2 contingency table has df = (3 − 1)(2 − 1) = 2, and at α = 0.05 the critical value is 5.991.

Chi-Square Distribution Critical Values

Each cell is the value of χ² with the column's probability in the upper tail. For example, with df = 10, a chi-square variable exceeds 18.307 with probability 0.05 and exceeds 3.940 with probability 0.95.

Critical values of the chi-square distribution by degrees of freedom and upper-tail area
df	upper-tail area (probability to the right of the value)
	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
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

Degrees of Freedom for Common Procedures

Goodness-of-fit test: df = k − 1, where k is the number of categories. Subtract one more degree for each parameter estimated from the data.
Test of independence: df = (r − 1)(c − 1) for an r×c contingency table. A 4×3 table has df = 6.
Single variance or standard deviation: df = n − 1, where n is the sample size.

To run the full test rather than look up cutoffs, use the chi-square calculator, which computes the statistic and the exact p-value from your observed and expected counts.

Using Both Tails: Variance Confidence Intervals

A 95% confidence interval for a population variance needs two critical values because the distribution is skewed. From a sample of n = 11 (df = 10), read the upper-tail 0.975 column (χ² = 3.247) and the upper-tail 0.025 column (χ² = 20.483). The interval is then:

(n − 1)s² / 20.483 ≤ σ² ≤ (n − 1)s² / 3.247

Note the reversal: the larger critical value produces the lower bound. This is the most common place the left-block columns of the table are used.

Shape of the Distribution

A chi-square variable with df degrees of freedom is the sum of df squared independent standard normal variables. Its mean is df and its variance is 2·df, so both the center and the spread of each row grow with the degrees of freedom — you can see the values shift right down every column. The distribution is strongly right-skewed for small df and becomes approximately normal for large df, which is why printed tables traditionally stop around df = 100.

The chi-square distribution also connects to the other reference tables on this site: it describes the squared values behind the standard normal table and forms the numerator and denominator of the F ratio in the F-table.

Frequently Asked Questions

How do I find a critical value in the chi-square table?

Find the row for your degrees of freedom and the column for your upper-tail area (the significance level for a standard test). The cell where they meet is the critical value. For a goodness-of-fit test with df = 3 at alpha = 0.05, the row for 3 and the column for 0.05 give 7.815: reject the null hypothesis when your chi-square statistic exceeds 7.815.

Why does the chi-square table only contain positive values?

The chi-square statistic is a sum of squared quantities, so it can never be negative. Its distribution starts at zero and stretches to the right with a long tail. That is also why the table is not symmetric like the t-table: the lower critical values (columns 0.995 to 0.90) must be printed separately instead of being obtained by flipping a sign.

Do the column headers refer to the left tail or the right tail?

The headers in this table are upper-tail (right-tail) areas: the probability that a chi-square variable exceeds the table value. A header of 0.05 means 5% of the distribution lies to the right of the printed value. Columns like 0.975 therefore give small numbers, because 97.5% of the distribution sits above them — these are the lower cutoffs used in confidence intervals. Some textbooks instead print cumulative (left-tail) probabilities; those headers are simply 1 minus these.

Why would I ever need the 0.995 to 0.90 columns?

Confidence intervals for a variance or standard deviation need both tails of the distribution. A 95% confidence interval for a variance with df = 10 uses the upper-tail 0.975 column (3.247) for one bound and the upper-tail 0.025 column (20.483) for the other. Hypothesis tests can also be left-tailed when the claim is that variability is below a target.

How do I work out the degrees of freedom for my test?

It depends on the procedure. A goodness-of-fit test over k categories uses df = k - 1 (minus one more for each parameter estimated from the data). A test of independence on an r-by-c contingency table uses df = (r - 1)(c - 1). A confidence interval or test for a single variance from a sample of n observations uses df = n - 1.

What if my degrees of freedom are not listed in the table?

Between df = 30 and df = 100 the table steps by 10; round down to the nearest listed row for a conservative critical value, or interpolate between neighboring rows for a closer estimate. For exact values at any degrees of freedom — including beyond 100 — use the critical value calculator on this site, which computes the same inverse chi-square function the table was built from.