Grubbs's test is based on the assumption of
normality. That is, one should first verify that the data can be reasonably approximated by a normal distribution before applying the Grubbs test. Grubbs's test detects one outlier at a time. This outlier is expunged from the dataset and the test is iterated until no outliers are detected. However, multiple iterations change the probabilities of detection, and the test should not be used for sample sizes of six or fewer since it frequently tags most of the points as outliers. Grubbs's test is defined for the following
hypotheses: :H0: There are no outliers in the data set :Ha: There is exactly one outlier in the data set The Grubbs
test statistic is defined as : G = \frac{\displaystyle\max_{i=1,\ldots, n}\left \vert Y_i - \bar{Y}\right\vert}{s} with n, \overline{Y}, and s denoting the number of measurements in the sample,
sample mean, and
standard deviation, respectively. The Grubbs test statistic is the largest
absolute deviation from the sample mean in units of the sample standard deviation. This is the
two-sided test, for which the hypothesis of no outliers is rejected at
significance level α if : G > \frac{n-1}{\sqrt{n}} \sqrt{\frac{t_{\alpha/(2n),n-2}^2}{n - 2 + t_{\alpha/(2n),n-2}^2}} with
tα/(2n),n−2 denoting the upper
critical value of the
t-distribution with
n − 2
degrees of freedom and a significance level of α/(2
n).
One-sided case Grubbs's test can also be defined as a one-sided test, replacing α/(2
n) with α/
n. To test whether the minimum value is an outlier, the test statistic is : G = \frac{\bar{Y}-Y_\min}{s} with
Ymin denoting the minimum value. To test whether the maximum value is an outlier, the test statistic is : G = \frac{Y_\max - \bar{Y}}{s} with
Ymax denoting the maximum value. ==Related techniques==