Dixon's Q test

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:

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 > Q_table, where Q_table 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:

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:

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:

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 = Q_table, so we conclude 0.167 is indeed an outlier. However, at 95% confidence, Q = 0.455 < 0.466 = Q_table 0.167 is not considered an outlier.

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 two-tailed Dixon's Q test.

Number of values:	3	4	5	6	7	8	9	10
Q_90%:	0.941	0.765	0.642	0.560	0.507	0.468	0.437	0.412
Q_95%:	0.970	0.829	0.710	0.625	0.568	0.526	0.493	0.466
Q_99%:	0.994	0.926	0.821	0.740	0.680	0.634	0.598	0.568

References

^ Halpern, Arthur M. "Experimental physical chemistry : a laboratory textbook." 3rd ed. / Arthur M. Halpern, George C. McBane. New York : W. H. Freeman, c2006 Library of Congress^{[permanent dead link‍]}

External links

Main page of GNU R's package 'outlier' includes 'dixon.test' function.
Dixon's test in Communications – use of Dixon's test in cognitive radio communications (by Shivanshu Shrivastava)

[1] Halpern, Arthur M. "Experimental physical chemistry : a laboratory textbook." 3rd ed. / Arthur M. Halpern, George C. McBane. New York : W. H. Freeman, c2006 Library of Congress^{[permanent dead link‍]}

[1]

Dixon's Q test

Contents

Example

Table

See also

References

Further reading

External links