# 16.6: Critical Values for Dixon's Q-Test

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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 Dixon’s Q-Test, each of which calculates a value for Qij 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

$Q_\text{exp} = Q_{10} = \frac {|\text{outlier's value} - \text{nearest value}|} {\text{largest value} - \text{smallest value}} \nonumber$

The suspected outlier is rejected if Qexp 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.

 $$\frac {\alpha \ce{->}} {n \ce{ v }}$$ 0.1 0.05 0.04 0.02 0.01 3 0.941 0.97 0.976 0.988 0.994 4 0.765 0.829 0.846 0.889 0.926 5 0.642 0.71 0.729 0.78 0.821 6 0.56 0.625 0.644 0.698 0.74 7 0.507 0.568 0.586 0.637 0.68 8 0.468 0.526 0.543 0.59 0.634 9 0.437 0.493 0.51 0.555 0.598 10 0.412 0.466 0.483 0.527 0.568

This page titled 16.6: Critical Values for Dixon's Q-Test is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by David Harvey.