Fundamentals of Statistics contains material of various lectures and courses of H. Lohninger on statistics, data analysis and chemometrics......click here for more.

Grubbs' Outlier Test

Grubbs' outlier test (Grubbs 1969 and Stefansky 1972 ) checks normally distributed data for outliers. This implies that one has to check whether the data show a normal distribution before applying the Grubbs test. The Grubbs test always checks the value which shows the largest absolute deviation from the mean. If an outlier has been identified and removed, the test must not be repeated without adapting the critical value.

The application of the test is quite simple and straightforward: one searches the maximum of the absolute differences between the values xi and the mean . The result is divided by the standard deviation of the sample. If the resulting test statistic g is greater than the critical value, the corresponding value can be regarded to be an outlier. An extract of the critical values is shown in the following table:

n gcrit
α=0.05
gcrit
α=0.01
  n gcrit
α=0.05
gcrit
α=0.01
  n gcrit
α=0.05
gcrit
α=0.01
31.15431.1547  152.54832.8061 803.30613.6729
41.48121.4962  162.58572.8521 903.34773.7163
51.71501.7637  172.62002.8940 1003.38413.7540
61.88711.9728  182.65162.9325 1203.44513.8167
72.02002.1391  192.68092.9680 1403.49513.8673
82.12662.2744  202.70823.0008 1603.53733.9097
92.21502.3868  252.82173.1353 1803.57363.9460
102.29002.4821  302.90853.2361 2003.60553.9777
112.35472.5641  403.03613.3807 3003.72364.0935
122.41162.6357  503.12823.4825 4003.80324.1707
132.46202.6990  603.19973.5599 5003.86314.2283
142.50732.7554  703.25763.6217 6003.91094.2740

 

There is a one-sided alternative which allows to test either the minimum xmin or the maximum xmax of the entire data set. The test statistics calculates according to the following formulas:

A value can be regarded an outlier if the statistic g is greater than the critical value. Please note that in the case of the one-sided test the critical values are different. An extract is given below:

n gcrit
α=0.05
gcrit
α=0.01
  n gcrit
α=0.05
gcrit
α=0.01
  n gcrit
α=0.05
gcrit
α=0.01
31.15311.1546  152.40902.7049  803.13193.5208
41.46251.4925  162.44332.7470  903.17333.5632
51.67141.7489  172.47482.7854  1003.20953.6002
61.82211.9442  182.50402.8208  1203.27063.6619
71.93812.0973  192.53122.8535  1403.32083.7121
82.03172.2208  202.55662.8838  1603.36333.7542
92.10962.3231  252.66293.0086  1803.40013.7904
102.17612.4097  302.74513.1029  2003.43243.8220
112.23392.4843  402.86753.2395  3003.55253.9385
122.28502.5494  502.95703.3366  4003.63394.0166
132.33052.6070  603.02693.4111  5003.69524.0749
142.37172.6585  703.08393.4710  6003.74424.1214

Last Update: 2012-10-08