Dixon’s test

As before, the distribution Is not Gaussian and Dixon s test concludes that the maximum value is not realistic. 

This distribution belongs to a standard model, Dixon s test does not give rise to an unrealistic value (hence the normality). The calculation of the confidence range of the individual values in this sequence then leads to ... 

In the simplest case, if all sample sub groups have the same size, zzi = n2 =. .. = zzjt, Dixon s test for outliers can be used (see Sect. 4.3.2). Then, in Eq. (4.35), instead of the individual values, the means are entered. 

Sometimes, a value within a set might appear aberrant (this is known as an outlier). Although it might be tempting to reject this data point, it must be remembered that a value can only be aberrant relative to some law of probability. There is a simple statistical criterion on which to base the decision of whether to retain or reject this value. Dixon s test is based on the following ratio (as long as there are at least seven measurements) ... 

Table 21.4—Abridged table of critical values of Q (Dixon s test).

Dixon s test is a parametric test for detecting outliers. At least three results are needed. The dataset is arranged in order of increasing magnitude x1, x2...xn. Dixon s >-statistic is calculated. The equation depends on the number of results, g-statistic for three to seven observations ... 

TABLE 6.6. Critical Values for Dixon s Test. The Suspected Value is an Outlier if the Calculated 2-Statistic is Above the Critical Value... 

Applying the first method, four different criteria, namely Dixon s test, Grubbs test, the coefficient of dewness test and the coefficient of kurtosis test are used at a significance level of a = 0.05. If a laboratory mean for each element as single unweighted value was declared to be an outlier by any criterion, it is rejected and the whole procedure repeated until no more outliers could be identified. The remaining laboratory means are then combined in the usual way to provide estimates of the overall mean (consensus value) and its associated standard deviation, standard error and 95% confidence interval. 

Before using QC data, an appropriate statistical test, such as Grubb s or Dixon s tests, should be applied to test for outliers. Those data points acquired during a period in which the method was not in statistical control should not be included in the calculations. This approach assumes that measurements are being made at concentrations where the relative uncertainty is constant over a defined range, the constant uncertainty that would dominate at concentrations close to the limit of detection or limit of quantification is negligible, and that recovery is independent of concentration. 

To perform Dixon s test, we start by calculating the smallest and largest differences between the suspect element and the rest of the values of the data set. Then we take the ratio of the smallest difference to the largest, and compare the result with a tabulated critical Q-value that depends on the desired confidence level and the total niunber of elements in the sample. If the calculated ratio is larger than the tabulated value, we can consider the suspect value as an outlier. In oiu example, there exists only one suspect value (the 56.3 min. time obtained in the ninth experiment), so we will use the Q-values for a one-sided tail test. Since the extreme... 

The calculated ratio is much larger than the two tabulated values for Dixon s test. This result shows, as we already suspected, that the time of experiment 9 is really different from the others. In fact, we already knew that the conclusion would be this one, since the weather on that experiment s day was completely at3 ical. 

Testing for an outlier under the assumption of normal distribution can be carried out by the Dixon s test. This test uses the range of measurements and can be applied even in cases where only few data are available. The n measurements are arranged in ascending order. If the very small value to be tested as an outlier is denoted by Xi and the very large striking value by x, then the test statistics is calculated by... 

Table 4 Critical values of Q (P=0.05) for a two-sided Dixon s test for outliers...

Dixon s test (sometimes called the Q-test) is another test for outliers which is popular because the calculation is simple. For small samples (size 3 to 7) the test assesses a suspect measurement by comparing the difference between it and the... 

In order to use Dixon s test for an outlier, that is to test Hq all measurements come from the same population, the statistic Q is calculated ... 

Apply Dixon s test to the data from the previous example. 

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