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## Outlier TestDean and Dixon

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 x1 being the suspect value. Then the test statistic Q is calculated using the equation The decision whether x1 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 r10; for 8 <= N <=10 use r11; for 11 <= N <= 13 use r21, and for n >= 14 use r22:    The following tables show the critical values for r11, r21, and r22, respectively. r10 is equal to Q, its critical values can be obtained from the table above.

 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 106 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