Grubbs outlier test

The Grubbs outlier test is a statistical test used to outliers in a given sample to discover, eliminate and improve by iteration the remaining sample. The Nalimov outlier test is a variant of the Grubbs' outlier test.

requirements

The prerequisite for using the Grubbs test is the hypothesis that the data are approximately normally distributed . This can be derived from theoretical considerations or checked by a test for normal distribution for the source of the data in general or the sample in particular. The test should be carried out without the probable outliers, as these could otherwise falsify the test statistics. ${\displaystyle X_{1},\dotsc ,X_{n}}$

Test statistics

The test statistic is then derived from the sample

${\displaystyle G={\frac {\max {|X_{i}-{\bar {X}}|}}{S_{n}}}}$

calculated in the

${\displaystyle {\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}}$denote the sample mean and the corrected sample variance of the sample. Under the null hypothesis that there are no outliers, has a distribution that can be represented as a function of a t distribution . Consequently, the test for the significance level is rejected, if with ${\displaystyle S_{n}^{2}={\frac {1}{n-1}}\sum _{i=1}^{n}(X_{i}-{\bar {X}})^{2}}$ ${\displaystyle G}$ ${\displaystyle \alpha \;}$ ${\displaystyle G>z_{\alpha }}$

${\displaystyle z_{\alpha }={\frac {n-1}{\sqrt {n}}}{\sqrt {\frac {t_{{\frac {\alpha }{2n}},n-2}^{2}}{n-2+t_{{\frac {\alpha }{2n}},n-2}^{2}}}}}$

applies. Here referred to the - quantile of a distribution with degrees of freedom to the level . ${\displaystyle t_{\beta ,m}}$ ${\displaystyle \beta }$ ${\displaystyle t}$ ${\displaystyle m}$ ${\displaystyle \beta }$

If the test actually detects an outlier, the date in question is removed from the sample and a new Grubbs test is carried out with data in the remainder of the sample. This should continue until the test can no longer detect any outliers. ${\displaystyle n-1}$

variants

• A variant of the Grubbs test is to only want to discover outliers upwards or only those downwards. This is achieved by looking at or instead of only and adjusting the critical value of the test statistic from to . ${\displaystyle \max {|X_{i}-{\bar {X}}|}}$ ${\displaystyle \max X_{i}-{\bar {X}}}$ ${\displaystyle {\bar {X}}-\min X_{i}}$ ${\displaystyle t_{{\frac {\alpha }{2n}},n-2}}$ ${\displaystyle t_{{\frac {\alpha }{n}},n-2}}$
• The outlier test according to Nalimov modifies the test variable with a correction factor that depends on the sample size: ${\displaystyle G}$

${\displaystyle G={\frac {\max {|X_{i}-{\bar {X}}|}}{S_{n}}}{\sqrt {\frac {n}{n-1}}}}$.

literature

• Frank E. Grubbs: Sample Criteria for Testing Outlying Observations . In: The Annals of Mathematical Statistics . tape 21 , no. 1 , 1950, p. 27-58 , doi : 10.1214 / aoms / 1177729885 .
• Frank E. Grubbs: Procedures for Detecting Outlying Observations in Samples . In: Technometrics . tape 11 , no. 1 , January 1, 1969, p. 1-21 , doi : 10.2307 / 1266761 .
• Chapter 1.3.5.17. Grubbs' Test for Outliers . In: Carroll Croarkin, Paul Tobias (Eds.): NIST / SEMATECH e-Handbook of Statistical Methods .
• DIN 53 804 Part 1, Statistical evaluations - Measurable (continuous) characteristics. Section 8.2 outlier test according to Grubbs, September 1981
• DIN ISO 5725 Part 2, Accuracy (trueness and precision) of measuring methods and measuring results - A fundamental method for determining the repeatability and comparability of specified measuring methods. Paragraph 14.4 Grubbs' test, draft February 1991

Web links

• Online calculator from Graphpad