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Table of Contents Statistical Tests Outlier Tests Dean and Dixon Test | |
See also: survey on statistical tests, Outlier Tests, Distribution Calculator, Walsh's Outlier Test |
A test for outliers of normally distributed data which is particularly simple to apply has been developed by J.W. Dixon. In order to perform this test for outliers, the data set containing N values has to be sorted either in an ascending or descending order, with x_{1} being the suspect value. Then the test statistic Q is calculated using the equation
The decision whether x_{1} is an outlier is performed by comparing the value Q to the critical values listed in the following table:
N | a=0.001 | a=0.002 | a=0.005 | a=0.01 | a=0.02 | a=0.05 | a=0.1 | a=0.2 |
3 | 0.999 | 0.998 | 0.994 | 0.988 | 0.976 | 0.941 | 0.886 | 0.782 |
4 | 0.964 | 0.949 | 0.921 | 0.889 | 0.847 | 0.766 | 0.679 | 0.561 |
5 | 0.895 | 0.869 | 0.824 | 0.782 | 0.729 | 0.643 | 0.559 | 0.452 |
6 | 0.822 | 0.792 | 0.744 | 0.698 | 0.646 | 0.563 | 0.484 | 0.387 |
7 | 0.763 | 0.731 | 0.681 | 0.636 | 0.587 | 0.507 | 0.433 | 0.344 |
8 | 0.716 | 0.682 | 0.633 | 0.591 | 0.542 | 0.467 | 0.398 | 0.314 |
9 | 0.675 | 0.644 | 0.596 | 0.555 | 0.508 | 0.436 | 0.370 | 0.291 |
10 | 0.647 | 0.614 | 0.568 | 0.527 | 0.482 | 0.412 | 0.349 | 0.274 |
15 | 0.544 | 0.515 | 0.473 | 0.438 | 0.398 | 0.338 | 0.284 | 0.220 |
20 | 0.491 | 0.464 | 0.426 | 0.393 | 0.356 | 0.300 | 0.251 | 0.193 |
25 | 0.455 | 0.430 | 0.395 | 0.364 | 0.329 | 0.277 | 0.230 | 0.176 |
30 | 0.430 | 0.407 | 0.371 | 0.342 | 0.310 | 0.260 | 0.216 | 0.165 |
where N is the number of values and a is the level of significance.
Please note that Dean and Dixon suggested in a later paper to take a more elaborate approach by using different formulas for different sample sizes in order to avoid the problem of two outliers on the same side of the distribution. They defined the following ratios and recommended that the various ratios be applied as follows: for 3 <= N <=7 use r_{10}; for 8 <= N <=10 use r_{11}; for 11 <= N <= 13 use r_{21}, and for n >= 14 use r_{22}:
N | a=0.001 | a=0.002 | a=0.005 | a=0.01 | a=0.02 | a=0.05 | a=0.1 | a=0.2 |
8 | 0.799 | 0.769 | 0.724 | 0.682 | 0.633 | 0.554 | 0.480 | 0.386 |
9 | 0.750 | 0.720 | 0.675 | 0.634 | 0.586 | 0.512 | 0.441 | 0.352 |
10 | 0.713 | 0.683 | 0.637 | 0.597 | 0.551 | 0.477 | 0.409 | 0.325 |
N | a=0.001 | a=0.002 | a=0.005 | a=0.01 | a=0.02 | a=0.05 | a=0.1 | a=0.2 |
11 | 0.770 | 0.746 | 0.708 | 0.674 | 0.636 | 0.575 | 0.518 | 0.445 |
12 | 0.739 | 0.714 | 0.676 | 0.643 | 0.605 | 0.546 | 0.489 | 0.420 |
13 | 0.713 | 0.687 | 0.649 | 0.617 | 0.580 | 0.522 | 0.467 | 0.399 |
N | a=0.001 | a=0.002 | a=0.005 | a=0.01 | a=0.02 | a=0.05 | a=0.1 | a=0.2 |
14 | 0.732 | 0.708 | 0.672 | 0.640 | 0.603 | 0.546 | 0.491 | 0.422 |
15 | 0.708 | 0.685 | 0.648 | 0.617 | 0.582 | 0.524 | 0.470 | 0.403 |
16 | 0.691 | 0.667 | 0.630 | 0.598 | 0.562 | 0.505 | 0.453 | 0.386 |
17 | 0.671 | 0.647 | 0.611 | 0.580 | 0.545 | 0.489 | 0.437 | 0.373 |
18 | 0.652 | 0.628 | 0.594 | 0.564 | 0.529 | 0.475 | 0.424 | 0.361 |
19 | 0.640 | 0.617 | 0.581 | 0.551 | 0.517 | 0.462 | 0.412 | 0.349 |
20 | 0.627 | 0.604 | 0.568 | 0.538 | 0.503 | 0.450 | 0.401 | 0.339 |
25 | 0.574 | 0.550 | 0.517 | 0.489 | 0.457 | 0.406 | 0.359 | 0.302 |
30 | 0.539 | 0.517 | 0.484 | 0.456 | 0.425 | 0.376 | 0.332 | 0.278 |
35 | 0.511 | 0.490 | 0.459 | 0.431 | 0.400 | 0.354 | 0.311 | 0.260 |
40 | 0.490 | 0.469 | 0.438 | 0.412 | 0.382 | 0.337 | 0.295 | 0.246 |
45 | 0.475 | 0.454 | 0.423 | 0.397 | 0.368 | 0.323 | 0.283 | 0.234 |
50 | 0.460 | 0.439 | 0.410 | 0.384 | 0.355 | 0.312 | 0.272 | 0.226 |
60 | 0.437 | 0.417 | 0.388 | 0.363 | 0.336 | 0.294 | 0.256 | 0.211 |
70 | 0.422 | 0.403 | 0.374 | 0.349 | 0.321 | 0.280 | 0.244 | 0.201 |
80 | 0.408 | 0.389 | 0.360 | 0.337 | 0.310 | 0.270 | 0.234 | 0.192 |
90 | 0.397 | 0.377 | 0.350 | 0.326 | 0.300 | 0.261 | 0.226 | 0.185 |
100 | 0.387 | 0.368 | 0.341 | 0.317 | 0.292 | 0.253 | 0.219 | 0.179 |
Hint: | Please note that the critical values listed in the tables above have been calculated by performing 10^{6} random experiments per value. These values differ slightly from values published by various authors, many of them using interpolation techniques to estimate the critical values. |
Last Update: 2005-Mai-08