Multiple Comparisons and Outliers
Grubbs' Test Table: Critical Values for Detecting an Outlier
Critical values for Grubbs' test, which checks whether the most extreme value in a normally distributed sample is an outlier.
- n = 3 to 500
- One- and two-sided
- Test your data
Grubbs' Test Table
Rows: sample size n · Columns: α for a two-sided or one-sided test · Cells: critical G. The value is an outlier if G > the cell.
| Two-sided test (largest or smallest) | One-sided test | |||||
|---|---|---|---|---|---|---|
| n | α = 0.10 | α = 0.05 | α = 0.01 | α = 0.05 | α = 0.025 | α = 0.01 |
| 3 | 1.1531 | 1.1543 | 1.1547 | 1.1531 | 1.1543 | 1.1546 |
| 4 | 1.4625 | 1.4813 | 1.4962 | 1.4625 | 1.4813 | 1.4925 |
| 5 | 1.6714 | 1.7150 | 1.7637 | 1.6714 | 1.7150 | 1.7489 |
| 6 | 1.8221 | 1.8871 | 1.9728 | 1.8221 | 1.8871 | 1.9442 |
| 7 | 1.9381 | 2.0200 | 2.1391 | 1.9381 | 2.0200 | 2.0973 |
| 8 | 2.0317 | 2.1266 | 2.2744 | 2.0317 | 2.1266 | 2.2208 |
| 9 | 2.1096 | 2.2150 | 2.3868 | 2.1096 | 2.2150 | 2.3231 |
| 10 | 2.1761 | 2.2900 | 2.4821 | 2.1761 | 2.2900 | 2.4097 |
| 11 | 2.2339 | 2.3547 | 2.5641 | 2.2339 | 2.3547 | 2.4843 |
| 12 | 2.2850 | 2.4116 | 2.6357 | 2.2850 | 2.4116 | 2.5494 |
| 13 | 2.3305 | 2.4620 | 2.6990 | 2.3305 | 2.4620 | 2.6070 |
| 14 | 2.3717 | 2.5073 | 2.7554 | 2.3717 | 2.5073 | 2.6585 |
| 15 | 2.4090 | 2.5483 | 2.8061 | 2.4090 | 2.5483 | 2.7049 |
| 16 | 2.4433 | 2.5857 | 2.8521 | 2.4433 | 2.5857 | 2.7470 |
| 17 | 2.4748 | 2.6200 | 2.8940 | 2.4748 | 2.6200 | 2.7854 |
| 18 | 2.5040 | 2.6516 | 2.9325 | 2.5040 | 2.6516 | 2.8208 |
| 19 | 2.5312 | 2.6809 | 2.9680 | 2.5312 | 2.6809 | 2.8535 |
| 20 | 2.5566 | 2.7082 | 3.0008 | 2.5566 | 2.7082 | 2.8838 |
| 21 | 2.5804 | 2.7338 | 3.0314 | 2.5804 | 2.7338 | 2.9121 |
| 22 | 2.6028 | 2.7577 | 3.0599 | 2.6028 | 2.7577 | 2.9385 |
| 23 | 2.6239 | 2.7803 | 3.0866 | 2.6239 | 2.7803 | 2.9633 |
| 24 | 2.6439 | 2.8016 | 3.1117 | 2.6439 | 2.8016 | 2.9866 |
| 25 | 2.6629 | 2.8217 | 3.1353 | 2.6629 | 2.8217 | 3.0086 |
| 26 | 2.6809 | 2.8408 | 3.1577 | 2.6809 | 2.8408 | 3.0295 |
| 27 | 2.6981 | 2.8589 | 3.1788 | 2.6981 | 2.8589 | 3.0492 |
| 28 | 2.7145 | 2.8762 | 3.1989 | 2.7145 | 2.8762 | 3.0680 |
| 29 | 2.7301 | 2.8927 | 3.2179 | 2.7301 | 2.8927 | 3.0859 |
| 30 | 2.7451 | 2.9085 | 3.2361 | 2.7451 | 2.9085 | 3.1029 |
| 35 | 2.8118 | 2.9782 | 3.3156 | 2.8118 | 2.9782 | 3.1778 |
| 40 | 2.8675 | 3.0361 | 3.3807 | 2.8675 | 3.0361 | 3.2395 |
| 45 | 2.9153 | 3.0854 | 3.4354 | 2.9153 | 3.0854 | 3.2916 |
| 50 | 2.9570 | 3.1282 | 3.4825 | 2.9570 | 3.1282 | 3.3366 |
| 60 | 3.0269 | 3.1997 | 3.5598 | 3.0269 | 3.1997 | 3.4111 |
| 70 | 3.0839 | 3.2576 | 3.6217 | 3.0839 | 3.2576 | 3.4710 |
| 80 | 3.1319 | 3.3061 | 3.6729 | 3.1319 | 3.3061 | 3.5208 |
| 90 | 3.1733 | 3.3477 | 3.7164 | 3.1733 | 3.3477 | 3.5633 |
| 100 | 3.2095 | 3.3841 | 3.7540 | 3.2095 | 3.3841 | 3.6002 |
| 120 | 3.2706 | 3.4451 | 3.8166 | 3.2706 | 3.4451 | 3.6619 |
| 150 | 3.3429 | 3.5170 | 3.8894 | 3.3429 | 3.5170 | 3.7340 |
| 200 | 3.4324 | 3.6055 | 3.9777 | 3.4324 | 3.6055 | 3.8221 |
| 300 | 3.5524 | 3.7236 | 4.0935 | 3.5524 | 3.7236 | 3.9385 |
| 500 | 3.6952 | 3.8631 | 4.2283 | 3.6952 | 3.8631 | 4.0749 |
Tip: click any value to highlight its row and column.
How to read this table
- Compute the mean x̄ and standard deviation s of all n values, including the suspect value.
- Compute G = |xsuspect − x̄| / s for the value farthest from the mean.
- Use the two-sided columns if the outlier could be either the largest or the smallest value, and the one-sided columns if you decided in advance to test only one end.
- The value is an outlier if G is greater than the table value.
Worked example
Twelve assay results have mean 10.25 and SD 0.767, and the largest value, 12.6, gives G = (12.6 − 10.25) / 0.767 = 3.06. Row n = 12, two-sided α = 0.05 gives 2.4116. Because 3.06 > 2.4116, 12.6 is an outlier at the 5% level. Paste these values into the calculator above to see the full calculation.
Click the underlined links to highlight the value in the table.
Calculator
For values between the rows and columns of the table. Results update as you type.
Formula and how the values were computed
Critical values use the exact relationship with the t-distribution: G = ((n − 1)/√n) √(t² / (n − 2 + t²)), where t is the upper α/(2n) (two-sided) or α/n (one-sided) critical value with n − 2 df.
Frequently asked questions
Can I apply Grubbs' test repeatedly?
It is designed to detect one outlier. Repeating it after removing a value is common but can miss outliers that mask each other; the generalized ESD test handles several suspected outliers properly.
Should I delete a value that Grubbs' test flags?
Not automatically. A flagged value should be investigated for data entry or measurement errors. In clinical trials, outliers are normally kept in the primary analysis and handled with a pre-specified sensitivity analysis.