# Grubbs-Test für Ausreißer

Grubbs' test for outliers allows checking if the maximum or minimum of a sample is an outlier. It is available in Excel using the XLSTAT software.

## What is the Grubbs test

Grubbs (1950, 1969, 1972) developed several tests in order to determine whether the greatest value or the lowest value (Grubbs test) are outliers, or, for the double Grubbs test, whether the two greatest values or the two lowest ones are outliers. This test assumes that the data corresponds to a sample extracted from a population that follows a normal distribution.

## Detecting outliers

In statistics, an outlier is a value recorded for a given variable that seems unusual and suspiciously lower or greater than the other observed values. One can distinguish two types of outliers:

• An outlier can simply be related to a reading error (on an measuring instrument), a keyboarding error, or a special event that disrupted the observed phenomenon to the point of making it incomparable to others. In such cases, you must either correct the outlier, if possible, or otherwise remove the observation to avoid that it disturbs the analyses that are planed (descriptive analysis, modeling, predicting).
• An outlier can also be due to an atypical event, but nevertheless known or interesting to study. For example, if we study the presence of certain bacteria in river water, you can have samples without bacteria, and other with aggregates with many bacteria. These data are of course important to keep. The models used should reflect that potential dispersion.

When there are outliers in the data, depending on the stage of the study, we must identify them, possibly with the aid of tests, flag them in the reports (in tables or on graphical representations), delete or use methods able to treat them as such.

To identify outliers, there are different approaches. For example, in classical linear regression, we can use the value of Cook’s d values, or submit the standardized residuals to a Grubbs test to see if one or two values are abnormal. The classical Grubbs test can help identifying one outlier, while the double Grubbs test allows identifying two. It is not recommended to use these methods repeatedly on the same sample. However, it may be appropriate if you really suspect that there are more than two outliers.

## Results with XLSTAT

The results correspond to the Grubbs test are displayed. An interpretation of the test is provided if a single iteration of the test was requested, or if no observation was identified as being an outlier.
In case several iterations were required, also display a table showing, for each observation, the iteration in which it was removed from the sample.

The z-scores are displayed if they have been requested.

## References

Barnett V. and Lewis T. (1980). Outliers in Statistical Data. John Wiley and Sons, Chichester, New York, Brisbane, Toronto.

Grubbs F.E. (1950). Sample criteria for testing outlying observations. Ann. Math. Stat. 21, 27-58.

Grubbs F.E. (1969). Procedures for detecting outlying observations in samples. Technometrics,  11(1), 1-21.

Grubbs, F.E. and Beck G. (1972). Extension of sample sizes and percentage points for significance tests of outlying observations. Technometrics, 14, 847-854.

Hawkins D.M. (1980). Identification of Outliers. Chapman and Hall, London.

International Organization for Standardization (1994). ISO 5725-2: Accuracy (trueness and precision) of measurement methods and results—Part 2: Basic method for the determination of repeatability and reproducibility of a standard measurement method, Geneva.

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