14.1.6.9.4: Rejection of Outliers (Q-test)

In a set of collected data, there may be one or more values that deviate markedly from the trend of the others; in other words, there may be one or more data points that seem suspicious. If a measurement appears suspicious, we need evidence before deciding that it is statistically discordant with the rest of the data. One tool for doing this is the Q-test.

You should not reject a piece of data that seems suspect until you perform a Q-test.

In a set of data of three to ten measurements, the Q-test can be performed on a single suspicious data point as follows:
\[Q_{exp} \equiv \frac{\mid(\text { suspected outlier's value })-(\text {value closest to it }) \mid}{\text { (highest value })-(\text { lowest value })}\]

The following table provides critical values for \(Q_{critical} (\alpha, n)\), where:

• \(\alpha\) is the probability of incorrectly rejecting the suspected outlier,
• \(n\) is the number of samples in the data set.

Table \(\PageIndex{1}\): Critical Values for Dixon's Q-Test

\(\frac {\alpha \ce{->}} {\text{no. of samples} \ce{ v }}\)	\(Q_{critical}\) \(\alpha =0.1\) (90% confidence limit)	\(Q_{critical}\) \(\alpha =0.05\) (95% confidence limit)	\(Q_{critical}\) \(\alpha =0.04\) (96% confidence limit)	\(Q_{critical}\) \(\alpha =0.02\) (98% confidence limit)	\(Q_{critical}\) \(\alpha =0.01\) (99% confidence limit)
n = 3	0.941	0.970	0.976	0.988	0.994
n = 4	0.765	0.829	0.846	0.889	0.926
n = 5	0.642	0.710	0.729	0.780	0.821
n = 6	0.560	0.625	0.644	0.698	0.740
n = 7	0.507	0.568	0.586	0.637	0.680
n = 8	0.468	0.526	0.543	0.590	0.634
n = 9	0.437	0.493	0.510	0.555	0.598
n = 10	0.412	0.466	0.483	0.527	0.568

The Q-test:

Compare the value of \(Q_{exp}\) from the equation above to the critical value of \(Q_{critical}\).

• If \(Q_{exp} \ge Q_{critical}\) then the suspect value can be rejected.
• If \(Q_{exp} \le Q_{critical}\),the value should be retained in the data set.

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For additional information consult Rorabacher, D. B. “Statistical Treatment for Rejection of Deviant Values: Critical Values of Dixon’s ‘Q’ Parameter and Related Subrange Ratios at the 95% confidence Level,” Anal. Chem. 1991, 63, 139–146. (see also http://chem-net.blogspot.com/2013/01...atistical.html)