# Dixon's Q test

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In statistics, Dixon's Q test, or simply the Q test, is used for identification and rejection of outliers. This assumes normal distribution and per Robert Dean and Wilfrid Dixon, and others, this test should be used sparingly and never more than once in a data set. To apply a Q test for bad data, arrange the data in order of increasing values and calculate Q as defined:

${\displaystyle Q={\frac {\text{gap}}{\text{range}}}}$

Where gap is the absolute difference between the outlier in question and the closest number to it. If Q > Qtable, where Qtable is a reference value corresponding to the sample size and confidence level, then reject the questionable point. Note that only one point may be rejected from a data set using a Q test.

## Example

Consider the data set:

${\displaystyle 0.189,\ 0.167,\ 0.187,\ 0.183,\ 0.186,\ 0.182,\ 0.181,\ 0.184,\ 0.181,\ 0.177\,}$

Now rearrange in increasing order:

${\displaystyle 0.167,\ 0.177,\ 0.181,\ 0.181,\ 0.182,\ 0.183,\ 0.184,\ 0.186,\ 0.187,\ 0.189\,}$

We hypothesize that 0.167 is an outlier. Calculate Q:

${\displaystyle Q={\frac {\text{gap}}{\text{range}}}={\frac {0.177-0.167}{0.189-0.167}}=0.455.}$

With 10 observations and at 90% confidence, Q = 0.455 > 0.412 = Qtable, so we conclude 0.167 is an outlier. However, at 95% confidence, Q = 0.455 < 0.466 = Qtable 0.167 is not considered an outlier. This means that for this example we can be 90% sure that 0.167 is an outlier, but we cannot be 95% sure.

McBane[1] notes: Dixon provided related tests intended to search for more than one outlier, but they are much less frequently used than the r10 or Q version that is intended to eliminate a single outlier.

## Table

This table summarizes the limit values of the test.

 Number of values: 3 4 5 6 7 8 9 10 Q90%: 0.941 0.765 0.642 0.56 0.507 0.468 0.437 0.412 Q95%: 0.97 0.829 0.71 0.625 0.568 0.526 0.493 0.466 Q99%: 0.994 0.926 0.821 0.74 0.68 0.634 0.598 0.568