In
statistics, Dixon's ''Q'' test, or simply the ''Q'' test, is used for identification and rejection of
outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s. 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:
:
Where ''gap'' is the
absolute difference
The absolute difference of two real numbers x and y is given by , x-y, , the absolute value of their difference. It describes the distance on the real line between the points corresponding to x and y. It is a special case of the Lp distance for ...
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 that 0.167 is an outlier. Calculate ''Q'':
:
With 10 observations and at 90%
confidence
Confidence is a state of being clear-headed either that a hypothesis or prediction is correct or that a chosen course of action is the best or most effective. Confidence comes from a Latin word 'fidere' which means "to trust"; therefore, having ...
, ''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
[Halpern, Arthur M. "Experimental physical chemistry : a laboratory textbook." 3rd ed. / Arthur M. Halpern , George C. McBane. New York : W. H. Freeman, c200]
Library of Congress
/ref> 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.
See also
*Grubbs's test for outliers In statistics, Grubbs's test or the Grubbs test (named after Frank E. Grubbs, who published the test in 1950), also known as the maximum normalized residual test or extreme studentized deviate test, is a test used to detect outliers in a univariat ...
References
Further reading
* Robert B. Dean and Wilfrid J. Dixon (1951) "Simplified Statistics for Small Numbers of Observations". Anal. Chem., 1951, 23 (4), 636–638
AbstractFull 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, 200
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)
*W. J. Dixon. The Annals of Mathematical Statistics. Vol. 21, No. 4 (Dec., 1950), pp. 488-506
External links
includes 'dixon.test' function.
Dixon's test in Communications
– use of Dixon's test in cognitive radio communications (by Shivanshu Shrivastava)
{{DEFAULTSORT:Dixon's Q Test
Statistical tests
Robust statistics
Statistical outliers