Dixon's Q test

In statistics, Dixon's Q test, or simply the Q test, is used for identification and rejection of outliers. 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_calculated > Q_table then reject the questionable point.

Example

For the data:

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

Arranged in increasing order:

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

Outlier is 0.167. Calculate Q:

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

With 10 observations, Q_calculated (0.455) > Q_table (0.412), so reject it with 90% confidence. However, at 95% confidence, Q_calculated (0.455) < Q_table (0.466).

Therefore keep 0.167 at 95% confidence or reject it at 90% confidence.

Table

This table summarize 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

• 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)

External links

• Test for Outliers Main page of GNU R's package 'outlier' including 'dixon.test' function.