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 Dean and 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 an outlier. However, at 95% confidence, Q = 0.455 < 0.466 = Q_table 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	0.560	0.507	0.468	0.437	0.412
Q95%:	0.970	0.829	0.710	0.625	0.568	0.526	0.493	0.466
Q99%:	0.994	0.926	0.821	0.740	0.680	0.634	0.598	0.568

See also

• Grubbs' test for outliers

References

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

• R. B. Dean and W. J. Dixon (1951) "Simplified Statistics for Small Numbers of Observations". Anal. Chem., 1951, 23 (4), 636–638. Abstract Full text PDF
• Rorabacher, D.B. (1991) "Statistical Treatment for Rejection of Deviant Values: Critical Values of Dixon Q Parameter and Related Subrange Ratios at the 95 percent Confidence Level". Anal. Chem., 63 (2), 139–146. PDF (including larger tables of limit values)
• McBane, George C. (2006) "Programs to Compute Distribution Functions and Critical Values for Extreme Value Ratios for Outlier Detection". J. Statistical Software 16(3):1–9, 2006 Article (PDF) and Software (Fortan-90, Zipfile)
• Shivanshu Shrivastava, A. Rajesh, P. K. Bora (2014) "Sliding window Dixon's tests for malicious users' suppression in a cooperative spectrum sensing system" IET Communications, 2014, 8 (7)

External links

• Test for Outliers Main page of GNU R's package 'outlier' including 'dixon.test' function.
• Dixon's test in Communications Use of Dixon's test in Cognitive Radio Communications by Shivanshu Shrivastava