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 DeanDixon 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 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 x_{1} 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 r_{10}; for 8 <= N <=10 use r_{11}; for 11 <= N <= 13 use r_{21}, and for n >= 14 use r_{22}:
The following tables show the critical values for r_{11}, r_{21}, and r_{22}, respectively. r_{10} is equal to Q, its critical values can be obtained from the table above.
Critical values for r_{11}
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 
Critical values for r_{21}
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 
Critical values for r_{22}
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 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. 
