Variances and Regression Residuals

Cochran's C Test Table: Critical Values for the Largest Variance

Critical values for Cochran's test of whether the largest of several group variances is too large.

k = 2 to 120 groups
α = 0.05 and 0.01
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Cochran's C Table

Jump to a significance level below. Rows: df per group (n − 1) · Columns: number of groups k · Cells: critical C. Reject if C > the cell.

α = 0.05

α = 0.05
Number of groups (k) → df (n − 1) ↓	2	3	4	5	6	7	8	9	10	11	12	15	20	30	40	60	120
1	0.9985	0.9669	0.9065	0.8413	0.7807	0.7270	0.6798	0.6385	0.6020	0.5697	0.5410	0.4709	0.3894	0.2929	0.2369	0.1737	0.0998
2	0.9750	0.8709	0.7679	0.6838	0.6161	0.5612	0.5157	0.4775	0.4450	0.4169	0.3924	0.3346	0.2705	0.1979	0.1575	0.1132	0.0633
3	0.9392	0.7977	0.6839	0.5981	0.5321	0.4800	0.4377	0.4027	0.3733	0.3482	0.3264	0.2758	0.2205	0.1593	0.1258	0.0895	0.0494
4	0.9057	0.7457	0.6287	0.5440	0.4803	0.4307	0.3910	0.3584	0.3311	0.3080	0.2880	0.2419	0.1921	0.1377	0.1082	0.0765	0.0419
5	0.8772	0.7070	0.5894	0.5063	0.4447	0.3972	0.3594	0.3285	0.3028	0.2811	0.2624	0.2195	0.1735	0.1236	0.0968	0.0682	0.0371
6	0.8534	0.6770	0.5598	0.4783	0.4184	0.3726	0.3362	0.3067	0.2823	0.2616	0.2440	0.2034	0.1602	0.1137	0.0887	0.0623	0.0337
7	0.8332	0.6531	0.5365	0.4564	0.3980	0.3536	0.3185	0.2901	0.2666	0.2468	0.2299	0.1912	0.1502	0.1061	0.0827	0.0579	0.0312
8	0.8159	0.6333	0.5175	0.4387	0.3817	0.3384	0.3043	0.2768	0.2541	0.2350	0.2187	0.1815	0.1422	0.1003	0.0779	0.0544	0.0292
9	0.8010	0.6167	0.5018	0.4241	0.3682	0.3259	0.2927	0.2659	0.2439	0.2254	0.2096	0.1737	0.1358	0.0955	0.0741	0.0517	0.0277
10	0.7880	0.6025	0.4884	0.4118	0.3568	0.3154	0.2829	0.2568	0.2353	0.2173	0.2020	0.1671	0.1305	0.0915	0.0709	0.0494	0.0264
16	0.7341	0.5466	0.4365	0.3645	0.3136	0.2756	0.2461	0.2226	0.2033	0.1872	0.1735	0.1427	0.1106	0.0769	0.0593	0.0410	0.0217
36	0.6601	0.4748	0.3719	0.3065	0.2612	0.2278	0.2022	0.1819	0.1654	0.1517	0.1402	0.1144	0.0878	0.0603	0.0461	0.0316	0.0164
144	0.5813	0.4031	0.3093	0.2513	0.2119	0.1833	0.1616	0.1446	0.1308	0.1195	0.1100	0.0889	0.0675	0.0458	0.0347	0.0235	0.0120
∞	0.5000	0.3333	0.2500	0.2000	0.1667	0.1429	0.1250	0.1111	0.1000	0.0909	0.0833	0.0667	0.0500	0.0333	0.0250	0.0167	0.0083

α = 0.01

α = 0.01
Number of groups (k) → df (n − 1) ↓	2	3	4	5	6	7	8	9	10	11	12	15	20	30	40	60	120
1	0.9999	0.9933	0.9676	0.9279	0.8828	0.8376	0.7945	0.7544	0.7175	0.6837	0.6528	0.5747	0.4799	0.3632	0.2940	0.2151	0.1225
2	0.9950	0.9423	0.8643	0.7885	0.7218	0.6644	0.6152	0.5727	0.5358	0.5036	0.4751	0.4069	0.3297	0.2412	0.1916	0.1371	0.0759
3	0.9794	0.8832	0.7814	0.6957	0.6258	0.5685	0.5210	0.4810	0.4469	0.4175	0.3919	0.3318	0.2654	0.1914	0.1507	0.1068	0.0583
4	0.9586	0.8335	0.7212	0.6329	0.5635	0.5080	0.4627	0.4251	0.3934	0.3663	0.3428	0.2882	0.2288	0.1635	0.1281	0.0902	0.0489
5	0.9373	0.7933	0.6761	0.5875	0.5195	0.4659	0.4227	0.3870	0.3572	0.3318	0.3099	0.2593	0.2048	0.1455	0.1135	0.0796	0.0429
6	0.9172	0.7606	0.6410	0.5531	0.4866	0.4347	0.3932	0.3592	0.3308	0.3068	0.2861	0.2386	0.1877	0.1327	0.1033	0.0722	0.0387
7	0.8988	0.7335	0.6129	0.5259	0.4609	0.4105	0.3705	0.3378	0.3106	0.2876	0.2680	0.2228	0.1748	0.1231	0.0956	0.0667	0.0356
8	0.8823	0.7107	0.5897	0.5038	0.4401	0.3911	0.3523	0.3207	0.2945	0.2725	0.2536	0.2104	0.1646	0.1156	0.0897	0.0624	0.0332
9	0.8674	0.6912	0.5702	0.4853	0.4229	0.3751	0.3373	0.3067	0.2814	0.2601	0.2419	0.2003	0.1564	0.1096	0.0848	0.0589	0.0313
10	0.8539	0.6743	0.5536	0.4697	0.4084	0.3617	0.3248	0.2950	0.2704	0.2497	0.2321	0.1919	0.1496	0.1046	0.0809	0.0561	0.0297
16	0.7949	0.6058	0.4882	0.4094	0.3529	0.3105	0.2774	0.2509	0.2292	0.2110	0.1956	0.1608	0.1244	0.0863	0.0664	0.0457	0.0240
36	0.7064	0.5152	0.4056	0.3349	0.2856	0.2492	0.2213	0.1990	0.1810	0.1660	0.1533	0.1250	0.0958	0.0657	0.0502	0.0342	0.0177
144	0.6063	0.4230	0.3252	0.2644	0.2230	0.1929	0.1701	0.1521	0.1376	0.1257	0.1157	0.0935	0.0709	0.0480	0.0364	0.0245	0.0125
∞	0.5000	0.3333	0.2500	0.2000	0.1667	0.1429	0.1250	0.1111	0.1000	0.0909	0.0833	0.0667	0.0500	0.0333	0.0250	0.0167	0.0083

Tip: click any value to highlight its row and column.

How to read this table

Compute the variance of each of the k groups (all of size n).
C = largest variance / sum of all the variances.
Find the row for df = n − 1 and the column for k.
The largest variance is significantly too large if C exceeds the table value.

Worked example

Five laboratories each run 6 replicates and report variances of 4.1, 3.6, 12.8, 5.0 and 4.4. C = 12.8 / 29.9 = 0.428. Row df = 5, column k = 5 gives 0.5063, so laboratory 3 is not significantly more variable than the others at the 5% level.

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Formula and how the values were computed

C = s²_max / Σ sᵢ² · C_crit = 1 / (1 + (k − 1)/F_{α/k; n−1, (k−1)(n−1)})

Values use Cochran's relationship with the F distribution, C = 1 / (1 + (k − 1)/F), where F is the upper α/k point of F on (n − 1) and (k − 1)(n − 1) df. This Bonferroni-type bound is the standard way the table is computed and is essentially exact at these significance levels.

Frequently asked questions

Where is Cochran's C used?

In interlaboratory studies (ISO 5725) to screen laboratories with unusually large within-laboratory variance, and as a check on homogeneity of variance before ANOVA.

Is this the same as Cochran's Q test?

No. Cochran's Q is a test for matched binary outcomes. Cochran's C, tabulated here, compares variances.

Cochran's C Test Table: Critical Values for the Largest Variance

Cochran's C Table

α = 0.05

α = 0.01

How to read this table

Worked example

Calculator

Formula and how the values were computed

Frequently asked questions

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