# statistical justification for rejection

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Q TEST FOR EXCLUSION OF DATA POINT(S)A question that often arises is the statistical justification for rejection of a divergentobservation. The question is not serious if enough data are at hand to establish aresonably valid estimate of standard deviation. The t test is available as a criterion and, inany event, the effect of a single divergent result on the mean value is relatively small. Forsmall groups of 3 to 8 replicates, however, the question is a more difficult one. Objectivecriteria for rejection or retention of a discrepant value from small numbers of observationsoften may be of the type that either rejects data too easily or retains highly suspect data.The Q test is one that is relatively critical and statistically sound. Q is defined as the ratio of thedivergence of the discordant value from its nearest neighbour to the range of values. If the valueof Q exceeds tabular values (see table), which depend on the number of observations, thequestionable value may be rejected. In the table, the tabular values of Q correspond to a 90%confidence limit that an error of the first kind, the rejection of valid measurements, has beenavoided. For small numbers of observations, say 3 to 5, the Q test allows rejection only of grosslydivergent values. It therefore increases the probability of an error of the second kind, the retentionof an erroneous result. The only valid justification for rejection of one discrepant observation froma group of 3 or 4 appears to be the location of an definable cause of determinant error. To rejectan intuitively doubtful observation is a dangerous practice, and the determination is best repeateduntil a statistically valid basis for retention or rejection is obtained. Otherwise the doubtful valueshould be retained.The median, though less efficient than the mean for observations with a Guassian distribution,often is recommended because of its insensitivity to a divergent value, especially when dealingwith small numbers of observations.Table: Q test values for rejection at the 90% confidence intervalCalculateQ = (pt. in question – nearest point)/(pt. in question – furthest point)RejectionIF Q > Q90%THEN it may be rejectedn 3 4 5 6Q90% 0.94 0.76 0.64 0.56n 7 8 9 10Q90% 0.51 0.47 0.44 0.41

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