# Dixon's Q test

In statistics, **Dixon's Q test**, or simply the

**, 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*Q*test for bad data, arrange the data in order of increasing values and calculate

*Q*as defined:

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:

Now rearrange in increasing order:

We hypothesize 0.167 is an outlier. Calculate *Q*:

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 r_{10} 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 |

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 |

## See also

## 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

- 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)

## External links

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