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Some simpler statistical tests for rejecting outliers in quantitative data*

Large data sets (N>100):

 

∑    

For small data sets:

∑ 1  

Rule of Huge Error: If you have a single outlier, then you can discard it with 98% confidence ifany of the following conditions are met.

| |  

5 8 6 

8 14 5 

15 4 

Dixon’s Q-test: If you have a single outlier, and your data has a normal distribution, then you candiscard the outlier if . Order the data values in increasing or decreasing order, such thatthe outlier is the final data point (x N).

3 7  

8 10  

11 13

 

*Compiled from:-Personal webpage of Prof. James K. Hardy, Dept. of Chemistry, University of Akron, “Statistical Treatment of Data” at

http://ull.chemistry.uakron.edu/analytical/Statistics/. This has good notes for basic statistics and refers to specific tests for the rejection of data and

discusses large and small sample sets.

-“Dixon's Q-test: Detection of a single outlier”, which includes an Applet for doing Q-test calculations and a brief discussion on rejecting data fromsmall data sets, on the University of Athen’s Department of Chemistry website at http://www.chem.uoa.gr/Applets/AppletQtest/Appl_Qtest2.html.

 Note: although much of the department’s website is in Greek, this page is in English.

-“Statistical Treatment of Analytical Data: Outliers (Chapter 6)” by Z.B. Alfassi, Z. Boger and Y. Ronen. CRC Press: 2005. This chapter is available

for reading through Google books if your library doesn’t have a copy.

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Grubbs’ T-test: This test can be used to evaluate multiple possible outliers. Start with the furthestoutlier, | | , and discard it if T > Tcrit.

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If you discard the outlier, and suspect others, then recalculate  and s in order to evaluate the next

furthest point.

Qcrit Values for Dixon's Q-test Outliers

Data points

N 0.5 1 5 10

3 0.994 0.988 0.941 0.886

4 0.926 0.889 0.765 0.679

5 0.821 0.780 0.642 0.557

6 0.740 0.698 0.560 0.482

7 0.680 0.637 0.507 0.434

8 0.725 0.683 0.554 0.479

9 0.677 0.635 0.512 0.441

10 0.639 0.597 0.477 0.409

11 0.713 0.679 0.576 0.517

12 0.675 0.642 0.546 0.490

13 0.649 0.615 0.521 0.467

Risk of false rejection (%)

Tcrit Values for Grubbs' T-test for Outliers

Data points

N 0.1 0.5 1 5 10

3 1.155 1.155 1.155 1.153 1.148

4 1.496 1.496 1.492 1.463 1.425

5 1.780 1.764 1.749 1.672 1.602

6 2.011 1.973 1.944 1.822 1.729

7 2.201 2.139 2.097 1.938 1.828

8 2.358 2.274 2.221 2.032 1.909

9 2.492 2.387 2.323 2.110 1.977

10 2.606 2.482 2.410 2.176 2.036

15 2.997 2.806 2.705 2.409 2.247

20 3.230 3.001 2.884 2.557 2.385

25 3.389 3.135 3.009 2.663 2.486

50 3.789 3.483 3.336 2.956 2.768

100 4.084 3.754 3.600 3.207 3.017

Risk of false rejection (%)