Grubbs' test (also known as the maximum normalized residual test or extreme studentized deviate test) is a statistical test used to detect outliers in a univariate dataset assumed to come from a normally distributed population. Grubbs' test is defined for the hypothesis:

• H_0: the dataset does not contain outliers.
• H_1: the dataset contains exactly one outlier.

The Grubbs' test statistic for a two-sided alternative hypothesis is defined as

where s is the sample standard deviation. The Grubbs test statistic is thus the largest absolute deviation from the sample mean in units of the sample standard deviation.

The Grubbs' test statistic for the alternative hypothesis that the minimum value is an outlier is defined as

The Grubbs' test statistic for the alternative hypothesis that the maximum value is an outlier is defined as

For a two-sided test, the hypothesis that a dataset does not contain an outlier is rejected at significance level α if

where t denotes the upper critical value of the t-distribution with N-2 degrees of freedom and a significance level of α/(2N).

For a one-sided test, the hypothesis that a dataset does not contain an outlier is rejected at significance level α if

where t denotes the upper critical value of the t-distribution with N-2 degrees of freedom and a significance level of α/N.

npm install @stdlib/stats-incr-grubbs

Alternatively,

• To load the package in a website via a script tag without installation and bundlers, use the ES Module available on the esm branch (see README).
• If you are using Deno, visit the deno branch (see README for usage intructions).
• For use in Observable, or in browser/node environments, use the Universal Module Definition (UMD) build available on the umd branch (see README).

The branches.md file summarizes the available branches and displays a diagram illustrating their relationships.

To view installation and usage instructions specific to each branch build, be sure to explicitly navigate to the respective README files on each branch, as linked to above.

var incrgrubbs = require( '@stdlib/stats-incr-grubbs' );

Returns an accumulator function which incrementally performs Grubbs' test for outliers.

var accumulator = incrgrubbs();

The function accepts the following options:

• alpha: significance level. Default: 0.05.

• alternative: alternative hypothesis. The option may be one of the following values:

  • 'two-sided': test whether the minimum or maximum value is an outlier.
  • 'min': test whether the minimum value is an outlier.
  • 'max': test whether the maximum value is an outlier.

  Default: 'two-sided'.

• init: number of data points the accumulator should use to compute initial statistics before testing for an outlier. Until the accumulator is provided the number of data points specified by this option, the accumulator returns null. Default: 100.

If provided an input value x, the accumulator function returns updated test results. If not provided an input value x, the accumulator function returns the current test results.

var rnorm = require( '@stdlib/random-base-normal' );

var opts = {
    'init': 0
};
var accumulator = incrgrubbs( opts );

var results = accumulator( rnorm( 10.0, 5.0 ) );
// returns null

results = accumulator( rnorm( 10.0, 5.0 ) );
// returns null

results = accumulator( rnorm( 10.0, 5.0 ) );
// returns <Object>

results = accumulator();
// returns <Object>

The accumulator function returns an object having the following fields:

• rejected: boolean indicating whether the null hypothesis should be rejected.
• alpha: significance level.
• criticalValue: critical value.
• statistic: test statistic.
• df: degrees of freedom.
• mean: sample mean.
• sd: corrected sample standard deviation.
• min: minimum value.
• max: maximum value.
• alt: alternative hypothesis.
• method: method name.
• print: method for pretty-printing test output.

The print method accepts the following options:

• digits: number of digits after the decimal point. Default: 4.
• decision: boolean indicating whether to print the test decision. Default: true.

• Grubbs' test assumes that data is normally distributed. Accordingly, one should first verify that the data can be reasonably approximated by a normal distribution before applying the Grubbs' test.
• The accumulator must be provided at least three data points before performing Grubbs' test. Until at least three data points are provided, the accumulator returns null.
• Input values are not type checked. If provided NaN or a value which, when used in computations, results in NaN, the test statistic is NaN for all future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly before passing the value to the accumulator function.

• Grubbs, Frank E. 1950. "Sample Criteria for Testing Outlying Observations." The Annals of Mathematical Statistics 21 (1). The Institute of Mathematical Statistics: 27–58. doi:10.1214/aoms/1177729885.
• Grubbs, Frank E. 1969. "Procedures for Detecting Outlying Observations in Samples." Technometrics 11 (1). Taylor & Francis: 1–21. doi:10.1080/00401706.1969.10490657.

• @stdlib/stats-incr/mgrubbs: moving Grubbs' test for outliers.

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