Variances and Regression Residuals
Mandel's k Statistic Table: Critical Values for Within-Laboratory Consistency
Critical values of Mandel's k, the ratio of each laboratory's within-laboratory SD to the pooled SD, used in ISO 5725-2 to find laboratories with unusually variable results.
- p = 3 to 100 laboratories
- n = 2 to 10 replicates
- α = 0.05 and 0.01
- Test your data
Mandel's k Table
Jump to a significance level below. Rows: number of laboratories p · Columns: replicates per laboratory n · Cells: critical k. A laboratory is flagged if k exceeds the cell.
α = 0.05
| Replicates per laboratory (n) → Laboratories (p) ↓ | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | n = 7 | n = 8 | n = 9 | n = 10 |
|---|---|---|---|---|---|---|---|---|---|
| 3 | 1.645 | 1.526 | 1.453 | 1.404 | 1.369 | 1.341 | 1.319 | 1.301 | 1.286 |
| 4 | 1.757 | 1.589 | 1.500 | 1.443 | 1.402 | 1.371 | 1.347 | 1.326 | 1.309 |
| 5 | 1.814 | 1.623 | 1.526 | 1.465 | 1.421 | 1.388 | 1.362 | 1.341 | 1.323 |
| 6 | 1.848 | 1.644 | 1.543 | 1.479 | 1.433 | 1.399 | 1.372 | 1.350 | 1.331 |
| 7 | 1.870 | 1.659 | 1.554 | 1.488 | 1.442 | 1.407 | 1.379 | 1.356 | 1.337 |
| 8 | 1.885 | 1.669 | 1.562 | 1.495 | 1.448 | 1.412 | 1.384 | 1.361 | 1.342 |
| 9 | 1.896 | 1.677 | 1.568 | 1.500 | 1.452 | 1.416 | 1.388 | 1.365 | 1.345 |
| 10 | 1.904 | 1.683 | 1.573 | 1.505 | 1.456 | 1.420 | 1.391 | 1.367 | 1.348 |
| 11 | 1.910 | 1.687 | 1.577 | 1.508 | 1.459 | 1.422 | 1.393 | 1.370 | 1.350 |
| 12 | 1.915 | 1.691 | 1.580 | 1.511 | 1.462 | 1.425 | 1.396 | 1.372 | 1.352 |
| 13 | 1.920 | 1.695 | 1.583 | 1.513 | 1.464 | 1.427 | 1.397 | 1.373 | 1.353 |
| 14 | 1.923 | 1.697 | 1.585 | 1.515 | 1.466 | 1.428 | 1.399 | 1.375 | 1.355 |
| 15 | 1.926 | 1.700 | 1.587 | 1.517 | 1.467 | 1.430 | 1.400 | 1.376 | 1.356 |
| 16 | 1.929 | 1.702 | 1.589 | 1.518 | 1.468 | 1.431 | 1.401 | 1.377 | 1.357 |
| 17 | 1.931 | 1.704 | 1.591 | 1.520 | 1.470 | 1.432 | 1.402 | 1.378 | 1.358 |
| 18 | 1.933 | 1.705 | 1.592 | 1.521 | 1.471 | 1.433 | 1.403 | 1.379 | 1.358 |
| 19 | 1.934 | 1.707 | 1.593 | 1.522 | 1.472 | 1.434 | 1.404 | 1.379 | 1.359 |
| 20 | 1.936 | 1.708 | 1.594 | 1.523 | 1.472 | 1.434 | 1.405 | 1.380 | 1.360 |
| 21 | 1.937 | 1.709 | 1.595 | 1.524 | 1.473 | 1.435 | 1.405 | 1.381 | 1.360 |
| 22 | 1.938 | 1.710 | 1.596 | 1.524 | 1.474 | 1.436 | 1.406 | 1.381 | 1.361 |
| 23 | 1.939 | 1.711 | 1.597 | 1.525 | 1.475 | 1.436 | 1.406 | 1.382 | 1.361 |
| 24 | 1.940 | 1.712 | 1.598 | 1.526 | 1.475 | 1.437 | 1.407 | 1.382 | 1.362 |
| 25 | 1.941 | 1.713 | 1.598 | 1.526 | 1.476 | 1.437 | 1.407 | 1.383 | 1.362 |
| 26 | 1.942 | 1.714 | 1.599 | 1.527 | 1.476 | 1.438 | 1.408 | 1.383 | 1.362 |
| 27 | 1.943 | 1.714 | 1.600 | 1.527 | 1.477 | 1.438 | 1.408 | 1.383 | 1.363 |
| 28 | 1.943 | 1.715 | 1.600 | 1.528 | 1.477 | 1.439 | 1.408 | 1.384 | 1.363 |
| 29 | 1.944 | 1.715 | 1.601 | 1.528 | 1.477 | 1.439 | 1.409 | 1.384 | 1.363 |
| 30 | 1.945 | 1.716 | 1.601 | 1.529 | 1.478 | 1.439 | 1.409 | 1.384 | 1.363 |
| 31 | 1.945 | 1.716 | 1.601 | 1.529 | 1.478 | 1.440 | 1.409 | 1.384 | 1.364 |
| 32 | 1.946 | 1.717 | 1.602 | 1.529 | 1.478 | 1.440 | 1.409 | 1.385 | 1.364 |
| 33 | 1.946 | 1.717 | 1.602 | 1.530 | 1.479 | 1.440 | 1.410 | 1.385 | 1.364 |
| 34 | 1.947 | 1.718 | 1.603 | 1.530 | 1.479 | 1.440 | 1.410 | 1.385 | 1.364 |
| 35 | 1.947 | 1.718 | 1.603 | 1.530 | 1.479 | 1.441 | 1.410 | 1.385 | 1.365 |
| 36 | 1.947 | 1.718 | 1.603 | 1.531 | 1.479 | 1.441 | 1.410 | 1.386 | 1.365 |
| 37 | 1.948 | 1.719 | 1.604 | 1.531 | 1.480 | 1.441 | 1.411 | 1.386 | 1.365 |
| 38 | 1.948 | 1.719 | 1.604 | 1.531 | 1.480 | 1.441 | 1.411 | 1.386 | 1.365 |
| 39 | 1.948 | 1.719 | 1.604 | 1.531 | 1.480 | 1.441 | 1.411 | 1.386 | 1.365 |
| 40 | 1.949 | 1.720 | 1.604 | 1.532 | 1.480 | 1.442 | 1.411 | 1.386 | 1.365 |
| 50 | 1.951 | 1.722 | 1.606 | 1.533 | 1.482 | 1.443 | 1.412 | 1.387 | 1.367 |
| 60 | 1.953 | 1.724 | 1.608 | 1.534 | 1.483 | 1.444 | 1.413 | 1.388 | 1.367 |
| 80 | 1.955 | 1.725 | 1.609 | 1.536 | 1.484 | 1.445 | 1.414 | 1.389 | 1.368 |
| 100 | 1.956 | 1.726 | 1.610 | 1.537 | 1.485 | 1.446 | 1.415 | 1.390 | 1.369 |
α = 0.01
| Replicates per laboratory (n) → Laboratories (p) ↓ | n = 2 | n = 3 | n = 4 | n = 5 | n = 6 | n = 7 | n = 8 | n = 9 | n = 10 |
|---|---|---|---|---|---|---|---|---|---|
| 3 | 1.715 | 1.643 | 1.578 | 1.528 | 1.488 | 1.456 | 1.430 | 1.408 | 1.389 |
| 4 | 1.917 | 1.772 | 1.673 | 1.604 | 1.553 | 1.513 | 1.481 | 1.454 | 1.431 |
| 5 | 2.051 | 1.849 | 1.729 | 1.649 | 1.591 | 1.546 | 1.511 | 1.481 | 1.456 |
| 6 | 2.142 | 1.900 | 1.766 | 1.679 | 1.616 | 1.568 | 1.530 | 1.499 | 1.473 |
| 7 | 2.207 | 1.937 | 1.793 | 1.700 | 1.634 | 1.584 | 1.544 | 1.511 | 1.484 |
| 8 | 2.256 | 1.964 | 1.812 | 1.716 | 1.647 | 1.595 | 1.554 | 1.521 | 1.493 |
| 9 | 2.294 | 1.985 | 1.827 | 1.728 | 1.657 | 1.604 | 1.562 | 1.528 | 1.499 |
| 10 | 2.324 | 2.001 | 1.839 | 1.737 | 1.665 | 1.611 | 1.569 | 1.534 | 1.505 |
| 11 | 2.348 | 2.015 | 1.849 | 1.745 | 1.672 | 1.617 | 1.574 | 1.539 | 1.509 |
| 12 | 2.368 | 2.026 | 1.857 | 1.752 | 1.678 | 1.622 | 1.578 | 1.542 | 1.513 |
| 13 | 2.385 | 2.035 | 1.864 | 1.757 | 1.682 | 1.626 | 1.582 | 1.546 | 1.516 |
| 14 | 2.399 | 2.044 | 1.870 | 1.762 | 1.686 | 1.629 | 1.585 | 1.549 | 1.518 |
| 15 | 2.411 | 2.051 | 1.875 | 1.766 | 1.690 | 1.632 | 1.588 | 1.551 | 1.521 |
| 16 | 2.422 | 2.057 | 1.879 | 1.769 | 1.693 | 1.635 | 1.590 | 1.553 | 1.523 |
| 17 | 2.431 | 2.062 | 1.883 | 1.773 | 1.695 | 1.637 | 1.592 | 1.555 | 1.524 |
| 18 | 2.440 | 2.067 | 1.887 | 1.775 | 1.698 | 1.639 | 1.594 | 1.557 | 1.526 |
| 19 | 2.447 | 2.071 | 1.890 | 1.778 | 1.700 | 1.641 | 1.595 | 1.558 | 1.527 |
| 20 | 2.454 | 2.075 | 1.893 | 1.780 | 1.702 | 1.643 | 1.597 | 1.559 | 1.528 |
| 21 | 2.460 | 2.078 | 1.895 | 1.782 | 1.703 | 1.644 | 1.598 | 1.561 | 1.529 |
| 22 | 2.465 | 2.081 | 1.897 | 1.784 | 1.705 | 1.646 | 1.599 | 1.562 | 1.530 |
| 23 | 2.470 | 2.084 | 1.899 | 1.785 | 1.706 | 1.647 | 1.601 | 1.563 | 1.531 |
| 24 | 2.475 | 2.087 | 1.901 | 1.787 | 1.708 | 1.648 | 1.602 | 1.564 | 1.532 |
| 25 | 2.479 | 2.089 | 1.903 | 1.788 | 1.709 | 1.649 | 1.602 | 1.565 | 1.533 |
| 26 | 2.483 | 2.091 | 1.905 | 1.790 | 1.710 | 1.650 | 1.603 | 1.565 | 1.534 |
| 27 | 2.486 | 2.093 | 1.906 | 1.791 | 1.711 | 1.651 | 1.604 | 1.566 | 1.534 |
| 28 | 2.490 | 2.095 | 1.908 | 1.792 | 1.712 | 1.652 | 1.605 | 1.567 | 1.535 |
| 29 | 2.493 | 2.097 | 1.909 | 1.793 | 1.713 | 1.653 | 1.606 | 1.567 | 1.536 |
| 30 | 2.496 | 2.099 | 1.910 | 1.794 | 1.713 | 1.653 | 1.606 | 1.568 | 1.536 |
| 31 | 2.498 | 2.100 | 1.911 | 1.795 | 1.714 | 1.654 | 1.607 | 1.568 | 1.537 |
| 32 | 2.501 | 2.102 | 1.912 | 1.796 | 1.715 | 1.655 | 1.607 | 1.569 | 1.537 |
| 33 | 2.503 | 2.103 | 1.913 | 1.797 | 1.716 | 1.655 | 1.608 | 1.569 | 1.538 |
| 34 | 2.505 | 2.104 | 1.914 | 1.797 | 1.716 | 1.656 | 1.608 | 1.570 | 1.538 |
| 35 | 2.507 | 2.106 | 1.915 | 1.798 | 1.717 | 1.656 | 1.609 | 1.570 | 1.538 |
| 36 | 2.509 | 2.107 | 1.916 | 1.799 | 1.717 | 1.657 | 1.609 | 1.571 | 1.539 |
| 37 | 2.511 | 2.108 | 1.917 | 1.799 | 1.718 | 1.657 | 1.610 | 1.571 | 1.539 |
| 38 | 2.513 | 2.109 | 1.917 | 1.800 | 1.718 | 1.658 | 1.610 | 1.571 | 1.539 |
| 39 | 2.514 | 2.110 | 1.918 | 1.800 | 1.719 | 1.658 | 1.610 | 1.572 | 1.540 |
| 40 | 2.516 | 2.111 | 1.919 | 1.801 | 1.719 | 1.658 | 1.611 | 1.572 | 1.540 |
| 50 | 2.528 | 2.118 | 1.924 | 1.805 | 1.723 | 1.662 | 1.614 | 1.575 | 1.542 |
| 60 | 2.536 | 2.123 | 1.927 | 1.808 | 1.725 | 1.664 | 1.615 | 1.576 | 1.544 |
| 80 | 2.546 | 2.128 | 1.932 | 1.811 | 1.728 | 1.666 | 1.618 | 1.578 | 1.546 |
| 100 | 2.552 | 2.132 | 1.934 | 1.814 | 1.730 | 1.668 | 1.619 | 1.580 | 1.547 |
Tip: click any value to highlight its row and column.
How to read this table
- Compute the within-laboratory standard deviation si for each of the p laboratories (each with n replicates).
- Compute ki = si / √(Σ sj² / p), the laboratory SD divided by the root mean square of all p SDs.
- Find the row for p and the column for n, in the 5% and 1% tables.
- ISO 5725-2 calls a laboratory a straggler if k exceeds the 5% value and an outlier if it exceeds the 1% value. Only large values are flagged (an unusually small SD is not).
Worked example
Six laboratories each measure a material n = 3 times and report within-laboratory SDs of 0.21, 0.18, 0.25, 0.52, 0.19, 0.23. The root mean square of the SDs is 0.2882. Laboratory 4 has the largest ratio, k = 0.52 / 0.2882 = 1.804.
For p = 6 and n = 3, the 5% value (Row p = 6, n = 3) is 1.644 and the 1% value (Row p = 6, n = 3) is 1.900. The value k = 1.804 is above the 5% value but below the 1% value, so it is a straggler.
Click the underlined links to highlight the value in the table.
Calculator
Paste the within-laboratory SD of each laboratory and the number of replicates.
Formula and how the values were computed
Under normality, k²/p follows a Beta((n − 1)/2, (p − 1)(n − 1)/2) distribution, so the critical value is exact: the square root of p times the upper α point of that beta distribution. α is the probability that one laboratory chosen in advance exceeds the value. As a check, for p = 3 and n = 2 the distribution of k²/3 is Beta(½, 1), so the upper 5% and 1% points are √(3 × 0.95²) = 1.645 and √(3 × 0.99²) = 1.715.
Frequently asked questions
What does Mandel's k measure?
It compares the repeatability SD of one laboratory with the typical SD across all laboratories. A large k means that laboratory's replicates are more spread out than the others'.
What is the difference between Mandel's k and Cochran's C?
Both flag a laboratory with a large within-laboratory variance. Cochran's C is a formal test using the largest variance only; Mandel's k is an indicator computed for every laboratory, usually plotted for all laboratories at once.
Why only upper critical values?
A small within-laboratory SD does not invalidate results in the same way as a large one, and ISO 5725-2 flags only the upper tail.
What if laboratories have different numbers of replicates?
The table assumes the same n for every laboratory. With unequal n, use the exact beta distribution in the calculator for each laboratory's own n as an approximation, or follow the unbalanced procedure in ISO 5725-2.