Multiple Comparisons and Outliers
Rosner's Generalized ESD Test Table: Critical Values for Detecting Several Outliers
Critical values λᵢ for the generalized extreme studentized deviate (ESD) test, which detects up to r outliers in a normally distributed sample without having to guess how many there are.
- n = 15 to 500
- Up to 10 outliers
- α = 0.05 and 0.01
- Test your data
Rosner's Generalized ESD Table
Jump to an overall significance level below. Rows: sample size n · Columns: test number i · Cells: critical value λi. Compare Ri with λi; the number of outliers is the largest i with Ri > λi.
Overall α = 0.05
| Test number (i) → Sample size (n) ↓ | i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | i = 6 | i = 7 | i = 8 | i = 9 | i = 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 15 | 2.5483 | 2.5073 | 2.4620 | 2.4116 | 2.3547 | 2.2900 | 2.2150 | 2.1266 | 2.0200 | 1.8871 |
| 20 | 2.7082 | 2.6809 | 2.6516 | 2.6200 | 2.5857 | 2.5483 | 2.5073 | 2.4620 | 2.4116 | 2.3547 |
| 25 | 2.8217 | 2.8016 | 2.7803 | 2.7577 | 2.7338 | 2.7082 | 2.6809 | 2.6516 | 2.6200 | 2.5857 |
| 30 | 2.9085 | 2.8927 | 2.8762 | 2.8589 | 2.8408 | 2.8217 | 2.8016 | 2.7803 | 2.7577 | 2.7338 |
| 35 | 2.9782 | 2.9653 | 2.9519 | 2.9380 | 2.9236 | 2.9085 | 2.8927 | 2.8762 | 2.8589 | 2.8408 |
| 40 | 3.0361 | 3.0253 | 3.0141 | 3.0026 | 2.9906 | 2.9782 | 2.9653 | 2.9519 | 2.9380 | 2.9236 |
| 45 | 3.0854 | 3.0761 | 3.0666 | 3.0567 | 3.0466 | 3.0361 | 3.0253 | 3.0141 | 3.0026 | 2.9906 |
| 50 | 3.1282 | 3.1201 | 3.1118 | 3.1032 | 3.0945 | 3.0854 | 3.0761 | 3.0666 | 3.0567 | 3.0466 |
| 60 | 3.1997 | 3.1932 | 3.1866 | 3.1799 | 3.1730 | 3.1660 | 3.1588 | 3.1514 | 3.1439 | 3.1362 |
| 70 | 3.2576 | 3.2523 | 3.2469 | 3.2413 | 3.2357 | 3.2300 | 3.2242 | 3.2182 | 3.2122 | 3.2060 |
| 80 | 3.3061 | 3.3016 | 3.2970 | 3.2924 | 3.2877 | 3.2829 | 3.2780 | 3.2730 | 3.2680 | 3.2628 |
| 90 | 3.3477 | 3.3438 | 3.3399 | 3.3359 | 3.3318 | 3.3277 | 3.3235 | 3.3192 | 3.3149 | 3.3106 |
| 100 | 3.3841 | 3.3807 | 3.3772 | 3.3737 | 3.3701 | 3.3665 | 3.3628 | 3.3591 | 3.3554 | 3.3516 |
| 150 | 3.5170 | 3.5149 | 3.5128 | 3.5106 | 3.5085 | 3.5063 | 3.5041 | 3.5019 | 3.4996 | 3.4974 |
| 200 | 3.6055 | 3.6040 | 3.6025 | 3.6010 | 3.5994 | 3.5979 | 3.5964 | 3.5948 | 3.5932 | 3.5916 |
| 300 | 3.7236 | 3.7226 | 3.7217 | 3.7207 | 3.7198 | 3.7188 | 3.7178 | 3.7169 | 3.7159 | 3.7149 |
| 500 | 3.8631 | 3.8626 | 3.8621 | 3.8615 | 3.8610 | 3.8605 | 3.8599 | 3.8594 | 3.8589 | 3.8583 |
Overall α = 0.01
| Test number (i) → Sample size (n) ↓ | i = 1 | i = 2 | i = 3 | i = 4 | i = 5 | i = 6 | i = 7 | i = 8 | i = 9 | i = 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| 15 | 2.8061 | 2.7554 | 2.6990 | 2.6357 | 2.5641 | 2.4821 | 2.3868 | 2.2744 | 2.1391 | 1.9728 |
| 20 | 3.0008 | 2.9680 | 2.9325 | 2.8940 | 2.8521 | 2.8061 | 2.7554 | 2.6990 | 2.6357 | 2.5641 |
| 25 | 3.1353 | 3.1117 | 3.0866 | 3.0599 | 3.0314 | 3.0008 | 2.9680 | 2.9325 | 2.8940 | 2.8521 |
| 30 | 3.2361 | 3.2179 | 3.1989 | 3.1788 | 3.1577 | 3.1353 | 3.1117 | 3.0866 | 3.0599 | 3.0314 |
| 35 | 3.3156 | 3.3010 | 3.2858 | 3.2700 | 3.2534 | 3.2361 | 3.2179 | 3.1989 | 3.1788 | 3.1577 |
| 40 | 3.3807 | 3.3686 | 3.3561 | 3.3431 | 3.3296 | 3.3156 | 3.3010 | 3.2858 | 3.2700 | 3.2534 |
| 45 | 3.4354 | 3.4252 | 3.4146 | 3.4037 | 3.3924 | 3.3807 | 3.3686 | 3.3561 | 3.3431 | 3.3296 |
| 50 | 3.4825 | 3.4736 | 3.4645 | 3.4551 | 3.4454 | 3.4354 | 3.4252 | 3.4146 | 3.4037 | 3.3924 |
| 60 | 3.5598 | 3.5529 | 3.5458 | 3.5386 | 3.5311 | 3.5235 | 3.5157 | 3.5077 | 3.4995 | 3.4911 |
| 70 | 3.6217 | 3.6161 | 3.6103 | 3.6044 | 3.5985 | 3.5924 | 3.5861 | 3.5798 | 3.5733 | 3.5666 |
| 80 | 3.6729 | 3.6682 | 3.6633 | 3.6585 | 3.6535 | 3.6484 | 3.6433 | 3.6380 | 3.6327 | 3.6272 |
| 90 | 3.7164 | 3.7123 | 3.7082 | 3.7040 | 3.6998 | 3.6955 | 3.6911 | 3.6866 | 3.6821 | 3.6775 |
| 100 | 3.7540 | 3.7505 | 3.7469 | 3.7432 | 3.7396 | 3.7358 | 3.7320 | 3.7282 | 3.7243 | 3.7204 |
| 150 | 3.8894 | 3.8872 | 3.8851 | 3.8829 | 3.8808 | 3.8786 | 3.8763 | 3.8741 | 3.8719 | 3.8696 |
| 200 | 3.9777 | 3.9762 | 3.9747 | 3.9732 | 3.9716 | 3.9701 | 3.9686 | 3.9670 | 3.9655 | 3.9639 |
| 300 | 4.0935 | 4.0926 | 4.0917 | 4.0908 | 4.0898 | 4.0889 | 4.0880 | 4.0870 | 4.0861 | 4.0851 |
| 500 | 4.2283 | 4.2278 | 4.2273 | 4.2268 | 4.2263 | 4.2258 | 4.2253 | 4.2247 | 4.2242 | 4.2237 |
Tip: click any value to highlight its row and column.
How to read this table
- Choose r, an upper limit on the number of suspected outliers. It is better to choose r too large than too small.
- For i = 1, 2, …, r: compute the mean and SD of the current sample, find the value farthest from the mean, and compute Ri = max|x − x̄| / s. Then remove that value and repeat on the smaller sample.
- Find the row for the original sample size n and the column for test number i. That cell is λi.
- The number of outliers is the largest i for which Ri > λi. All values removed up to and including step i are outliers.
Worked example
Twenty-five assay results (listed in the calculator below) are screened for up to r = 4 outliers at an overall α = 0.05. Each step removes the value farthest from the current mean and standard deviation: step 1 removes 13.4 with R1 = 3.287; step 2 removes 12.8 with R2 = 3.729; step 3 removes 11.9 with R3 = 4.163; step 4 removes 10.4 with R4 = 1.976.
The critical values come from the n = 25 row of the α = 0.05 table: λ1 = 2.8217, λ2 = 2.8016, λ3 = 2.7803, λ4 = 2.7577. The test uses the largest i with Ri > λi, here i = 3, so the 3 most extreme values (13.4, 12.8, 11.9) are outliers. Do not stop at the first step that fails to exceed its limit: an earlier R can fall below its limit while a later R exceeds its own.
Click the underlined links to highlight the value in the table.
Calculator
Paste your data, set r and the overall α. For sample sizes between the rows of the table.
Formula and how the values were computed
Values use Rosner's exact expression (Rosner, 1983), where tp, ν is the p-th percentile of Student's t with ν df. For i = 1 the formula is identical to the two-sided Grubbs' critical value, so the first column of this table agrees with the two-sided columns of the Grubbs' test table. The test is generally recommended for samples of about 25 or more values; smaller sample sizes are included for completeness.
Frequently asked questions
What is the difference between Rosner's test and Grubbs' test?
Grubbs' test checks for one outlier. Applying it repeatedly can miss outliers that hide each other (masking). The generalized ESD test checks for up to r outliers in one procedure, with the overall false-positive rate held at α.
How do I choose r?
Choose the largest number of outliers you think is plausible, not the number you expect. If r is larger than needed, the test still finds only the real outliers, because the answer is the largest i whose R exceeds its limit.
Why does the table start at n = 15?
Very small samples have too few degrees of freedom left once outliers are removed, and the test is generally recommended for about 25 or more values. Rows from n = 15 are included; use the calculator for other sample sizes, or Dixon's Q test for very small samples.
Does the data need to be normal?
Yes. The test assumes the data apart from the outliers come from a normal distribution. Check normality on the data with the suspected outliers removed.