Multiple Comparisons and Outliers

Dixon's Q Test Table: Critical Values for r₁₀, r₁₁, r₂₁ and r₂₂

Critical values for Dixon's gap-to-range tests for a single outlier in small samples.

n = 3 to 30
Four Dixon ratios
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Dixon's Q Test Table

Jump to a ratio below. Rows: sample size n · Columns: α for testing the more extreme of the two ends · Cells: critical ratio. Reject if the ratio ≥ the cell.

Ratio r₁₀ (Dixon's Q)

Ratio r₁₀ (Dixon's Q)
α → n ↓	0.2	0.1	0.05	0.02	0.01	0.005
3	0.886	0.941	0.970	0.988	0.994	0.997
4	0.679	0.765	0.830	0.889	0.921	0.943
5	0.558	0.642	0.710	0.781	0.823	0.858
6	0.484	0.562	0.628	0.698	0.743	0.781
7	0.434	0.507	0.569	0.637	0.681	0.719
8	0.398	0.467	0.526	0.591	0.633	0.671
9	0.370	0.436	0.492	0.555	0.596	0.633
10	0.348	0.412	0.466	0.526	0.566	0.602
11	0.331	0.392	0.444	0.503	0.541	0.576
12	0.316	0.375	0.426	0.483	0.521	0.555
13	0.304	0.361	0.410	0.466	0.503	0.536
14	0.293	0.349	0.397	0.451	0.487	0.520
15	0.284	0.338	0.385	0.438	0.474	0.506
16	0.275	0.329	0.375	0.427	0.462	0.493
17	0.268	0.321	0.366	0.417	0.451	0.482
18	0.262	0.313	0.358	0.408	0.442	0.472
19	0.256	0.307	0.350	0.400	0.433	0.463
20	0.250	0.300	0.343	0.392	0.425	0.455
21	0.245	0.295	0.337	0.386	0.418	0.447
22	0.241	0.290	0.331	0.379	0.411	0.440
23	0.237	0.285	0.326	0.373	0.405	0.433
24	0.233	0.280	0.321	0.368	0.399	0.428
25	0.229	0.276	0.317	0.363	0.394	0.422
26	0.226	0.272	0.312	0.358	0.389	0.417
27	0.223	0.269	0.309	0.354	0.384	0.412
28	0.220	0.266	0.305	0.350	0.380	0.408
29	0.217	0.262	0.301	0.346	0.376	0.403
30	0.214	0.259	0.298	0.342	0.372	0.399

Ratio r₁₁

Ratio r₁₁
α → n ↓	0.2	0.1	0.05	0.02	0.01	0.005
4	0.875	0.938	0.969	0.987	0.994	0.997
5	0.695	0.786	0.848	0.904	0.932	0.952
6	0.585	0.675	0.743	0.810	0.849	0.880
7	0.513	0.599	0.665	0.735	0.777	0.812
8	0.463	0.544	0.608	0.676	0.718	0.755
9	0.426	0.503	0.564	0.631	0.672	0.709
10	0.397	0.470	0.530	0.594	0.635	0.670
11	0.375	0.445	0.502	0.564	0.604	0.639
12	0.356	0.423	0.479	0.539	0.578	0.613
13	0.340	0.406	0.459	0.519	0.557	0.591
14	0.327	0.390	0.443	0.500	0.538	0.571
15	0.315	0.377	0.428	0.485	0.521	0.554
16	0.305	0.365	0.415	0.471	0.507	0.539
17	0.296	0.355	0.404	0.459	0.494	0.526
18	0.288	0.346	0.394	0.448	0.483	0.513
19	0.281	0.338	0.385	0.438	0.472	0.503
20	0.274	0.330	0.377	0.429	0.463	0.493
21	0.268	0.323	0.369	0.421	0.454	0.484
22	0.263	0.317	0.363	0.414	0.447	0.476
23	0.258	0.311	0.356	0.406	0.439	0.468
24	0.253	0.306	0.351	0.400	0.432	0.462
25	0.249	0.301	0.345	0.394	0.426	0.455
26	0.245	0.297	0.340	0.389	0.420	0.449
27	0.241	0.292	0.335	0.384	0.415	0.443
28	0.238	0.289	0.331	0.379	0.410	0.438
29	0.235	0.285	0.327	0.374	0.405	0.433
30	0.232	0.281	0.323	0.370	0.401	0.428

Ratio r₂₁

Ratio r₂₁
α → n ↓	0.2	0.1	0.05	0.02	0.01	0.005
5	0.951	0.976	0.988	0.995	0.998	0.999
6	0.820	0.876	0.913	0.946	0.962	0.973
7	0.717	0.781	0.829	0.875	0.901	0.922
8	0.642	0.708	0.758	0.810	0.841	0.866
9	0.587	0.651	0.702	0.755	0.788	0.816
10	0.545	0.607	0.657	0.710	0.744	0.773
11	0.511	0.572	0.621	0.674	0.707	0.736
12	0.484	0.543	0.591	0.643	0.676	0.705
13	0.461	0.519	0.565	0.616	0.649	0.678
14	0.442	0.498	0.544	0.594	0.626	0.655
15	0.425	0.480	0.525	0.574	0.606	0.635
16	0.411	0.464	0.509	0.557	0.589	0.617
17	0.398	0.451	0.494	0.542	0.573	0.601
18	0.387	0.439	0.481	0.528	0.559	0.587
19	0.377	0.428	0.470	0.516	0.547	0.574
20	0.367	0.418	0.459	0.505	0.535	0.562
21	0.359	0.409	0.450	0.495	0.525	0.552
22	0.352	0.401	0.441	0.486	0.516	0.542
23	0.345	0.393	0.433	0.477	0.507	0.533
24	0.338	0.386	0.426	0.470	0.499	0.525
25	0.332	0.379	0.419	0.462	0.491	0.517
26	0.327	0.373	0.412	0.456	0.484	0.510
27	0.322	0.368	0.406	0.449	0.478	0.503
28	0.317	0.363	0.401	0.444	0.472	0.497
29	0.312	0.358	0.396	0.438	0.466	0.491
30	0.308	0.353	0.391	0.433	0.461	0.485

Ratio r₂₂

Ratio r₂₂
α → n ↓	0.2	0.1	0.05	0.02	0.01	0.005
6	0.935	0.968	0.984	0.994	0.997	0.998
7	0.814	0.874	0.913	0.946	0.962	0.973
8	0.722	0.788	0.836	0.882	0.907	0.927
9	0.654	0.721	0.772	0.823	0.853	0.877
10	0.602	0.668	0.719	0.772	0.804	0.831
11	0.562	0.626	0.677	0.730	0.762	0.791
12	0.529	0.592	0.641	0.694	0.727	0.756
13	0.502	0.563	0.611	0.664	0.697	0.725
14	0.479	0.539	0.586	0.638	0.670	0.699
15	0.459	0.518	0.564	0.615	0.648	0.676
16	0.443	0.500	0.546	0.595	0.627	0.656
17	0.428	0.484	0.529	0.578	0.610	0.638
18	0.415	0.470	0.514	0.563	0.594	0.621
19	0.403	0.457	0.501	0.549	0.580	0.607
20	0.392	0.446	0.489	0.536	0.567	0.594
21	0.383	0.435	0.478	0.525	0.555	0.582
22	0.374	0.426	0.468	0.515	0.545	0.571
23	0.367	0.417	0.459	0.505	0.535	0.561
24	0.359	0.410	0.451	0.496	0.526	0.552
25	0.353	0.402	0.443	0.488	0.517	0.543
26	0.346	0.396	0.436	0.481	0.509	0.535
27	0.341	0.389	0.429	0.474	0.502	0.528
28	0.335	0.383	0.423	0.467	0.496	0.521
29	0.330	0.378	0.417	0.461	0.489	0.515
30	0.326	0.373	0.412	0.455	0.483	0.508

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

How to read this table

Sort the data. For the suspect value at either end, compute the gap to its nearest neighbour divided by the range: Q = r₁₀ = gap / range.
Dixon recommended r₁₀ for n = 3 to 7, r₁₁ for 8 to 10, r₂₁ for 11 to 13 and r₂₂ for 14 and above; all four are tabulated for every n.
Reject the suspect value if the ratio is greater than or equal to the table value.

Worked example

Eight replicate assays read 12.0, 12.1, 12.2, 12.2, 12.3, 12.3, 12.4 and 13.4. The gap is 13.4 − 12.4 = 1.0 and the range is 1.4, so Q = 0.714. Row 8, α = 0.05 of the r₁₀ table gives 0.526, so 13.4 is an outlier at the 5% level.

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Formula and how the values were computed

r₁₀ = (x₍ₙ₎ − x₍ₙ₋₁₎)/(x₍ₙ₎ − x₍₁₎) · r₁₁ = (x₍ₙ₎ − x₍ₙ₋₁₎)/(x₍ₙ₎ − x₍₂₎) · r₂₁ = (x₍ₙ₎ − x₍ₙ₋₂₎)/(x₍ₙ₎ − x₍₂₎) · r₂₂ = (x₍ₙ₎ − x₍ₙ₋₂₎)/(x₍ₙ₎ − x₍₃₎)

Critical values are percentiles of each ratio, taking the larger of the two ends, from 20 million simulated normal samples for each n; simulation error is below 0.001. They agree with Rorabacher's (1991) table for r₁₀ to within about 0.003, and for n = 3 they match the exact values 0.941, 0.970 and 0.994.

Frequently asked questions

Dixon's Q or Grubbs' test?

Grubbs' test uses the mean and SD and is generally more powerful for a single outlier. Dixon's ratios are simple to compute by hand and are common in analytical chemistry and small laboratory samples.

Why several ratios?

The r₁₁, r₂₁ and r₂₂ ratios skip the value next to the suspect one, so a second outlier at the same or the other end does not mask the first.

Dixon's Q Test Table: Critical Values for r₁₀, r₁₁, r₂₁ and r₂₂

Dixon's Q Test Table

Ratio r₁₀ (Dixon's Q)

Ratio r₁₁

Ratio r₂₁

Ratio r₂₂

How to read this table

Worked example

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Formula and how the values were computed

Frequently asked questions

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