Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more.  ## Outlier Test - Dean and Dixon

A test for outliers of normally distributed data which is particularly simple to apply has been developed by J.W. Dixon. This test is eminently suitable for small sample sizes; for samples having more than 30 observations the test for significance of Pearson and Hartley can be used as well. In order to perform the Dean-Dixon 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 is the number of observations, α is the level of significance): If the calculated value of Q is greater than the critical threshold the corresponding data value x1 is regarded to be an outlier.

 N α=0.001 α=0.002 α=0.005 α=0.01 α=0.02 α=0.05 α=0.1 α=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

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 α=0.001 α=0.002 α=0.005 α=0.01 α=0.02 α=0.05 α=0.1 α=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 α=0.001 α=0.002 α=0.005 α=0.01 α=0.02 α=0.05 α=0.1 α=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 α=0.001 α=0.002 α=0.005 α=0.01 α=0.02 α=0.05 α=0.1 α=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: 2012-10-08