3.5: Rejection of Outliers (Q-test)

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In a set of collected data, there may be one or more values that deviate markedly from the trend of the others. 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 the suspect data point:
\[Q_{exp} \equiv \frac{\mid(\text { suspected outlier's value })-(\text {value closest to it }) \mid}{\text { (highest value })-(\text { lowest value })}\]

Compare the value of \(Q_{exp}\) from the equation above to the critical value of \(Q_r\). If \(Q_{exp}\) is greater or equal to \(Q_r\) then the suspect value can be rejected. If \(Q_{exp}\) is less than \(Q_r\),the value should be retained in the data set.

The following table provides critical values for \(Q(\alpha, n)\), where \(\alpha\) is the probability of incorrectly rejecting the suspected outlier and \(n\) is the number of samples in the data set. There are several versions of the Q-Test, each of which calculates a value for Q_ij where i is the number of suspected outliers on one end of the data set and j is the number of suspected outliers on the opposite end of the data set. The critical values for Q here are for a single outlier, Q₁₀, where

\[Q_\text{exp} = Q_{10} = \frac{\mid(\text { suspected outlier's value })-(\text {value closest to it }) \mid}{\text { (highest value })-(\text { lowest value })} \nonumber \]

The suspected outlier is rejected if Q_exp is greater than \(Q(\alpha, n)\). 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.

Table \(\PageIndex{1}\): Critical Values for Dixon's Q-Test
\(\frac {\alpha \ce{->}} {n \ce{ v }}\)	0.1 (90% confidence limit)	0.05 (95% confidence limit)	0.04 (96% confidence limit)	0.02 (98% confidence limit)	0.01 (99% confidence limit)
3	0.941	0.970	0.976	0.988	0.994
4	0.765	0.829	0.846	0.889	0.926
5	0.642	0.710	0.729	0.780	0.821
6	0.560	0.625	0.644	0.698	0.740
7	0.507	0.568	0.586	0.637	0.680
8	0.468	0.526	0.543	0.590	0.634
9	0.437	0.493	0.510	0.555	0.598
10	0.412	0.466	0.483	0.527	0.568

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