In statistics, Grubbs's test or the Grubbs test (named after
Frank E. Grubbs, who published the test in 1950), also known as the maximum normalized
residual test or extreme studentized deviate test, is a
test
Test(s), testing, or TEST may refer to:
* Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities
Arts and entertainment
* ''Test'' (2013 film), an American film
* ''Test'' (2014 film) ...
used to detect
outlier
In statistics, an outlier is a data point that differs significantly from other observations. An outlier may be due to a variability in the measurement, an indication of novel data, or it may be the result of experimental error; the latter are ...
s in a
univariate
In mathematics, a univariate object is an expression (mathematics), expression, equation, function (mathematics), function or polynomial involving only one Variable (mathematics), variable. Objects involving more than one variable are ''wikt:multi ...
data set
A data set (or dataset) is a collection of data. In the case of tabular data, a data set corresponds to one or more table (database), database tables, where every column (database), column of a table represents a particular Variable (computer sci ...
assumed to come from a
normally distributed
In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is
f(x ...
population.
Definition
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
A hypothesis (: hypotheses) is a proposed explanation for a phenomenon. A scientific method, scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educ ...
:
:H
0: There are no outliers in the data set
:H
a: There is exactly one outlier in the data set
The Grubbs
test statistic
Test statistic is a quantity derived from the sample for statistical hypothesis testing.Berger, R. L.; Casella, G. (2001). ''Statistical Inference'', Duxbury Press, Second Edition (p.374) A hypothesis test is typically specified in terms of a tes ...
is defined as
:
with
and
denoting the
sample mean
The sample mean (sample average) or empirical mean (empirical average), and the sample covariance or empirical covariance are statistics computed from a sample of data on one or more random variables.
The sample mean is the average value (or me ...
and
standard deviation
In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
, respectively. The Grubbs test statistic is the largest
absolute deviation
In mathematics and statistics, deviation serves as a measure to quantify the disparity between an observed value of a variable and another designated value, frequently the mean of that variable. Deviations with respect to the sample mean and th ...
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
In statistical hypothesis testing, a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true. More precisely, a study's defined significance level, denoted by \alpha, is the ...
α if
:
with ''t''
α/(2''N''),''N''−2 denoting the upper
critical value Critical value or threshold value can refer to:
* A quantitative threshold in medicine, chemistry and physics
* Critical value (statistics), boundary of the acceptance region while testing a statistical hypothesis
* Value of a function at a crit ...
of the
t-distribution with ''N'' − 2
degrees of freedom
In many scientific fields, the degrees of freedom of a system is the number of parameters of the system that may vary independently. For example, a point in the plane has two degrees of freedom for translation: its two coordinates; a non-infinite ...
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
:
with ''Y''
min denoting the minimum value. To test whether the maximum value is an outlier, the test statistic is
:
with ''Y''
max denoting the maximum value.
Related techniques
Several
graphical technique
Statistical graphics, also known as statistical graphical techniques, are graphics used in the field of statistics for data visualization.
Overview
Whereas statistics and data analysis procedures generally yield their output in numeric or tabul ...
s can be used to detect outliers. A simple
run sequence plot
A run chart, also known as a run-sequence plot is a graph that displays observed data in a time sequence. Often, the data displayed represent some aspect of the output or performance of a manufacturing or other business process. It is therefore ...
, a
box plot
In descriptive statistics, a box plot or boxplot is a method for demonstrating graphically the locality, spread and skewness groups of numerical data through their quartiles.
In addition to the box on a box plot, there can be lines (which are ca ...
, or a
histogram
A histogram is a visual representation of the frequency distribution, distribution of quantitative data. To construct a histogram, the first step is to Data binning, "bin" (or "bucket") the range of values— divide the entire range of values in ...
should show any obviously outlying points. A
normal probability plot
The normal probability plot is a graphical technique to identify substantive departures from normality. This includes identifying outliers, skewness, kurtosis, a need for transformations, and mixtures. Normal probability plots are made of raw ...
may also be useful.
See also
*
Chauvenet's criterion
*
Peirce's criterion In robust statistics, Peirce's criterion is a rule for eliminating outliers from data sets, which was devised by Benjamin Peirce.
Outliers removed by Peirce's criterion
The problem of outliers
In data sets containing real-numbered measurements, ...
*
Q test
*
Studentized residual
In statistics, a studentized residual is the dimensionless ratio resulting from the division of a errors and residuals in statistics, residual by an estimator, estimate of its standard deviation, both expressed in the same Unit of measurement, ...
*
Tau distribution
Tau (; uppercase Τ, lowercase τ or \boldsymbol\tau; ) is the nineteenth letter of the Greek alphabet, representing the voiceless dental or alveolar plosive . In the system of Greek numerals, it has a value of 300.
The name in English is p ...
References
Further reading
*
*
{{NIST-PD
Statistical tests
Statistical outliers