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Grubbs test
Grubbs’ test can also be used to test for single outliers and is generally considered more robust than a Dixon test because:
- Dixon’s test compares the extreme value to a neighbouring value rather than the mean of the data set;
- Dixon’s test uses the range which is a less efficient measure of variability than the standard deviation.
Three Grubbs tests are available for testing apparent outliers in normally distributed data. As with the Dixon test, the data must first be arranged in ascending order.
The G’ test is for a single low (or high) value. The G’’ test is for a pair of values at opposite extremes of the data set and the G’’’ test is for a pair of low (or high) values on the same side of the data set.
In each case the calculated value is compared with the appropriate critical value. Some critical values for different Grubbs tests, at the 95% confidence level, are shown below.
For the G’ and G’’ tests the result of the test is significant and the outlying value(s) requires further investigation (see section on Handling outliers) if the calculated value exceeds the critical value.
For the G’’’ test the calculated value gets smaller as the suspected outliers get more extreme. The result of a G’’’ test is therefore significant if the calculated value is less than the critical value.
Critical value for the Grubbs test
(95% confidence level) |
n |
G’(1) |
G’’(2) |
G’’’(note)(3) |
3 |
1.154 |
1.993 |
- |
4 |
1.481 |
2.429 |
0.0002 |
5 |
1.715 |
2.755 |
0.0090 |
6 |
1.887 |
3.012 |
0.0349 |
7 |
2.020 |
3.222 |
0.0708 |
8 |
2.127 |
3.399 |
0.1101 |
9 |
2.215 |
3.552 |
0.1492 |
10 |
2.290 |
3.685 |
0.1864 |
15 |
2.548 |
4.173 |
0.3367 |
20 |
2.708 |
4.496 |
0.4391 |
Note: The critical values for G''' are lower critical values. A calculated value for G''' less than the critical value should therefore be considered statistically significant. |
(1)E. S. Pearson and C. C. Sekar, Biometrika, 1936, 28, 308
(2)H. A. David, H. O. Hartley and E. S. Pearson, Biometrika, 1954, 41, 482
(3)F. E. Grubbs and G. Beck, Technometrics, 1972, 14, 847
Example
Apply the Grubbs tests to determine whether there are any outliers (at the 95% confidence level) in the data from the interlaboratory study given in the section on the Dixon test.
G’
The critical value for a G’ test at the 95% confidence level with n = 7 is 2.020. The calculated value for Ghighest exceeds the critical value, so the highest value in the set would therefore be considered an outlier at the 95% confidence level.
When a single outlier is identified using the G’ test we would not normally carry out the additional tests until the cause of the outlier had been investigated. However, to illustrate the calculations the G’’ and G’’’ tests are shown below.
G’’
The critical value for a G’’ test at the 95% confidence level with n = 7 is 3.222. The calculated value is less than the critical value so the highest and the lowest value do not form a pair of outliers.
G’’’
The critical value for a G’’’ test at the 95% confidence level with n = 7 is 0.0708. Remember that for the G’’’ test a significant result is indicated if the calculated value is less than the critical value. In this case, both calculated values exceed the critical value so there is no evidence of a pair of high (or low) outliers in the data set.
Last modified on
18 August 2009.