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{{Short description|Statistical test}}
In statistics, '''Grubbs's test'''<!--possessive of the proper noun--> or the '''Grubbs test'''<!--attributive use of the proper name--> (named after [[Frank E. Grubbs]], who published the test in 1950<ref>{{cite journal |last=Grubbs |first=Frank E. |title=Sample criteria for testing outlying observations |journal=[[Annals of Mathematical Statistics]] |volume=21 |issue=1 |pages=27–58 |doi=10.1214/aoms/1177729885 |year=1950 |doi-access=free |hdl=2027.42/182780 |hdl-access=free }}</ref>), also known as the '''maximum normalized [[errors and residuals in statistics|residual]] test''' or '''extreme studentized deviate test''', is a [[Statistical hypothesis testing|test]] used to detect [[outlier]]s in a [[univariate]] data set assumed to come from a [[normal distribution|normally distributed]] population.

==Definition==
Grubbs's test is based on the assumption of [[normal distribution|normality]]. That is, one should first verify that the data can be reasonably approximated by a normal distribution before applying the Grubbs test.<ref>Quoted from the ''Engineering and Statistics Handbook'', paragraph 1.3.5.17, http://www.itl.nist.gov/div898/handbook/eda/section3/eda35h.htm</ref>

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.<ref>{{Cite journal|last1=Adikaram|first1=K. K. L. B.|last2=Hussein|first2=M. A.|last3=Effenberger|first3=M.|last4=Becker|first4=T.|date=2015-01-14|title=Data Transformation Technique to Improve the Outlier Detection Power of Grubbs's Test for Data Expected to Follow Linear Relation|journal=Journal of Applied Mathematics|volume=2015|pages=1–9|language=en|doi=10.1155/2015/708948|doi-access=free}}</ref>

Grubbs's test is defined for the following [[statistical hypothesis|hypotheses]]:
:H<sub>0</sub>: There are no outliers in the data set
:H<sub>a</sub>: There is exactly one outlier in the data set

The Grubbs test statistic is defined as
:<math>
G =  \frac{\displaystyle\max_{i=1,\ldots, N}\left \vert Y_i - \bar{Y}\right\vert}{s}
</math>
with <math>\overline{Y}</math> and <math>s</math> denoting the [[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

:<math>
G > \frac{N-1}{\sqrt{N}} \sqrt{\frac{t_{\alpha/(2N),N-2}^2}{N - 2 + t_{\alpha/(2N),N-2}^2}}
</math>

with ''t''<sub>α/(2''N''),''N''−2</sub> denoting the upper [[critical value (statistics)|critical value]] of the [[Student's t-distribution|t-distribution]] with ''N''&nbsp;−&nbsp;2 [[Degrees of freedom (statistics)|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
:<math>
G = \frac{\bar{Y}-Y_\min}{s}
</math>
with ''Y''<sub>min</sub> denoting the minimum value. To test whether the maximum value is an outlier, the test statistic is
:<math>
G = \frac{Y_\max - \bar{Y}}{s}
</math>
with ''Y''<sub>max</sub> denoting the maximum value.

==Related techniques==
Several [[graphical technique]]s can be used to detect outliers. A simple [[run sequence plot]], a [[box plot]], or a [[histogram]] should show any obviously outlying points. A [[normal probability plot]] may also be useful.

==See also==
* [[Chauvenet's criterion]]
* [[Peirce's criterion]]
* [[Q test]]
* [[Studentized residual]]
* [[Tau distribution]]

==References==
{{Reflist}}

==Further reading==
* {{cite journal|last=Grubbs|first=Frank|date=February 1969|title= Procedures for Detecting Outlying Observations in Samples|journal=Technometrics|volume=11|issue=1|pages=2–21|doi=10.2307/1266761|publisher=Technometrics, Vol. 11, No. 1|jstor=1266761}}
* {{cite journal|last=Stefansky|first=W.|year=1972|title=Rejecting Outliers in Factorial Designs|journal=Technometrics|pages=469–479|doi=10.2307/1267436|volume=14|issue=2|publisher=Technometrics, Vol. 14, No. 2|jstor=1267436}}
{{NIST-PD}}

[[Category:Statistical tests]]
[[Category:Statistical outliers]]