Multiple Comparisons and Outliers
Mandel's h Statistic Table: Critical Values for Between-Laboratory Consistency
Critical values of Mandel's h, the standardised deviation of each laboratory's mean from the mean of all laboratories, used in ISO 5725-2 to screen interlaboratory studies.
- p = 3 to 100 laboratories
- α = 0.10 to 0.005
- ISO 5725-2
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Mandel's h Table
Rows: number of laboratories p · Columns: significance level α · Cells: critical value of |h|. A laboratory is flagged if |h| exceeds the cell.
| Significance level (α) → Laboratories (p) ↓ | α = 0.1 | α = 0.05 | α = 0.025 | α = 0.01 | α = 0.005 |
|---|---|---|---|---|---|
| 3 | 1.140 | 1.151 | 1.154 | 1.155 | 1.155 |
| 4 | 1.350 | 1.425 | 1.462 | 1.485 | 1.492 |
| 5 | 1.441 | 1.571 | 1.652 | 1.715 | 1.742 |
| 6 | 1.489 | 1.656 | 1.772 | 1.872 | 1.922 |
| 7 | 1.518 | 1.711 | 1.852 | 1.983 | 2.054 |
| 8 | 1.538 | 1.749 | 1.909 | 2.065 | 2.152 |
| 9 | 1.553 | 1.777 | 1.952 | 2.127 | 2.229 |
| 10 | 1.563 | 1.798 | 1.985 | 2.176 | 2.290 |
| 11 | 1.572 | 1.815 | 2.011 | 2.215 | 2.339 |
| 12 | 1.579 | 1.829 | 2.032 | 2.248 | 2.380 |
| 13 | 1.585 | 1.840 | 2.050 | 2.275 | 2.415 |
| 14 | 1.590 | 1.850 | 2.065 | 2.298 | 2.444 |
| 15 | 1.594 | 1.858 | 2.078 | 2.318 | 2.469 |
| 16 | 1.597 | 1.865 | 2.089 | 2.335 | 2.491 |
| 17 | 1.600 | 1.871 | 2.099 | 2.350 | 2.511 |
| 18 | 1.603 | 1.876 | 2.107 | 2.363 | 2.528 |
| 19 | 1.605 | 1.881 | 2.115 | 2.375 | 2.543 |
| 20 | 1.607 | 1.885 | 2.121 | 2.385 | 2.557 |
| 21 | 1.609 | 1.889 | 2.127 | 2.395 | 2.569 |
| 22 | 1.611 | 1.893 | 2.133 | 2.403 | 2.580 |
| 23 | 1.613 | 1.896 | 2.138 | 2.411 | 2.590 |
| 24 | 1.614 | 1.899 | 2.142 | 2.418 | 2.600 |
| 25 | 1.615 | 1.901 | 2.147 | 2.425 | 2.608 |
| 26 | 1.617 | 1.904 | 2.150 | 2.431 | 2.616 |
| 27 | 1.618 | 1.906 | 2.154 | 2.436 | 2.623 |
| 28 | 1.619 | 1.908 | 2.157 | 2.442 | 2.630 |
| 29 | 1.620 | 1.910 | 2.160 | 2.446 | 2.636 |
| 30 | 1.620 | 1.911 | 2.163 | 2.451 | 2.642 |
| 31 | 1.621 | 1.913 | 2.166 | 2.455 | 2.647 |
| 32 | 1.622 | 1.915 | 2.168 | 2.459 | 2.653 |
| 33 | 1.623 | 1.916 | 2.170 | 2.463 | 2.657 |
| 34 | 1.623 | 1.917 | 2.173 | 2.466 | 2.662 |
| 35 | 1.624 | 1.919 | 2.175 | 2.469 | 2.666 |
| 36 | 1.625 | 1.920 | 2.177 | 2.472 | 2.670 |
| 37 | 1.625 | 1.921 | 2.178 | 2.475 | 2.674 |
| 38 | 1.626 | 1.922 | 2.180 | 2.478 | 2.677 |
| 39 | 1.626 | 1.923 | 2.182 | 2.480 | 2.681 |
| 40 | 1.627 | 1.924 | 2.183 | 2.483 | 2.684 |
| 50 | 1.630 | 1.931 | 2.195 | 2.502 | 2.709 |
| 60 | 1.633 | 1.936 | 2.203 | 2.514 | 2.726 |
| 80 | 1.636 | 1.942 | 2.213 | 2.530 | 2.746 |
| 100 | 1.638 | 1.946 | 2.219 | 2.539 | 2.758 |
Tip: click any value to highlight its row and column.
How to read this table
- Compute each laboratory's mean x̄i and the mean x̿ and SD s of these p means.
- For each laboratory compute hi = (x̄i − x̿) / s.
- Find the row for the number of laboratories p and read the 5% and 1% columns.
- ISO 5725-2 calls a laboratory a straggler if |h| exceeds the 5% value and an outlier if it exceeds the 1% value.
Worked example
Eight laboratories report mean results of 10.2, 10.4, 9.9, 10.1, 10.0, 10.3, 11.6, 9.4. The mean of the means is 10.238 and their SD is 0.630. Laboratory 7 has the largest deviation, h = (11.6 − 10.238) / 0.630 = 2.163.
For p = 8 laboratories, the 5% value (Row p = 8, α = 0.05) gives 1.749 and the 1% value (Row p = 8, α = 0.01) gives 2.065. Because |h| = 2.163 greater than 2.065, laboratory 7 is outlier (above the 1%% value).
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Formula and how the values were computed
Under normality, p·h²/(p − 1)² follows a Beta(½, (p − 2)/2) distribution, so the critical value is exact: (p − 1)/√p times the square root of the upper α point of that beta distribution. α is the probability that a single laboratory chosen in advance exceeds the value (two-sided). As a check, for p = 3 the distribution is an arcsine law and the 1% value, 1.155, equals the largest |h| that is possible for three laboratories, 2/√3.
Frequently asked questions
What is Mandel's h statistic?
It is a standardised measure of how far one laboratory's mean is from the average of all laboratories' means, in units of the SD of those means. Large positive or negative values suggest a laboratory is biased relative to the others.
What is the difference between a straggler and an outlier?
In ISO 5725-2 a straggler is a result flagged at the 5% level and an outlier is flagged at the 1% level. Stragglers are kept in the analysis; outliers are investigated and may be excluded if a cause is found.
Should I use Mandel's h or Grubbs' test?
They answer different questions. Mandel's h is a consistency check across laboratories, usually shown as a bar chart. Grubbs' test and Cochran's test are the formal outlier tests that ISO 5725-2 applies afterwards.
Is the p in the table the number of laboratories or the number of results?
The number of laboratories (cells) providing a mean for the same level of the material.