# Dixon Type (Q) Tests¶

Dixon Q-test algorithm uses the ratio of the gap between the possible a value outside the region of interest and the next closest value to it to the total range of the data set. This value is compared to a set of statistically determined values for a variety of confidence intervals. The basic formula is as follows:

$(1) \quad Q^{\text{test}} = \frac{Y_N - Y_{N-1}}{Y_N - Y_1} \, \text{,}$

and

$(2) \quad Q^{\text{test}} = \frac{Y_2 - Y_1}{Y_N - Y_1} \, \text{,}$

where (1) is used to test whether the largest value is a value outside the region of interest, and (2) is used to test whether the smallest value in the data set of length $$N$$ is a value outside the region of interest. If $$Q^{\text{test}} > Q^{\text{table}}$$, the value in question is a value outside the region of interest.

Input Parameters

Parameter Type Constraint Description Remarks
$$Y$$ $$Y \in \mathbb R^N$$ $$N \in \mathbb{N}$$ Input data sequenceof length $$N$$
$$Q^{\text{table}}$$     A table of calculated $$Q^{\text{table}}$$ values for a specific confidence level, e.g. $$90$$ %, $$95$$ % or $$99$$ %

Output Parameters

Parameter Type Constraint Description Remarks
$$\hat{Y}$$ $$\hat{Y} \in \mathbb R^N$$   Values in the $$Y$$ list which are outside the region of interest are marked

Single Steps using the Algorithm

References

• D.B. Rorabacher, 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., vol. 63(2), pp. 139-146, 1991.

http://pubs.acs.org/doi/pdf/10.1021/ac00002a010

• S. Walfish, A review of statistical outlier methods. Pharmaceutical Technology, 2006. Retrieved from www.pharmtech.com.