Statistics::Normality - test whether an empirical distribution can be taken as being drawn from a

       Mike Wendl, "<mwendl at genome.wustl.edu>"

       Please report any bugs or feature requests to "bug-statistics-normality at rt.cpan.org", or  through  the
       web   interface  at  <http://rt.cpan.org/NoAuth/ReportBug.html?Queue=Statistics-Normality>.   I  will  be
       notified, and then you'll automatically be notified of progress on your bug as I make changes.

       Copyright (C) 2011 Washington University

       This program is free software; you can redistribute it and/or modify  it  under  the  terms  of  the  GNU
       General  Public License as published by the Free Software Foundation; either version 2 of the License, or
       (at your option) any later version.

       This program is distributed in the hope that it will be useful, but WITHOUT ANY  WARRANTY;  without  even
       the  implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU General Public
       License for more details.

       You should have received a copy of the GNU General Public License along with this program; if not,  write
       to the Free Software Foundation, Inc., 59 Temple Place - Suite 330, Boston, MA 02111-1307, USA.

perl v5.34.0                                       2022-06-17                         Statistics::Normality(3pm)

       Various situations call for testing whether an empirical sample can be presumed to have been drawn from a
       normally (Gaussian <http://en.wikipedia.org/wiki/Normal_distribution>) distributed population, especially
       because many downstream significance tests depend upon the assumption of normality.  This package
       implements some of the more well-known tests <http://en.wikipedia.org/wiki/Normality_test> from the
       mathematical statistics literature, though there are also others that are not included.  The tests here
       are all so-called omnibus tests that find departures from normality on the basis of skewness and/or
       kurtosis [Dagostino71].  Note that, although the Kolmogorov-Smirnov test
       <http://en.wikipedia.org/wiki/Kolmogorov%E2%80%93Smirnov_test> can also be used in this capacity, it is a
       distance test and therefore not advisable [Dagostino71].  This, and other distance tests (e.g. Chi-
       square) are not implemented here.

       A list of functions that can be exported.  You can delete this section if you don't export anything, such
       as for a purely object-oriented module.

   Shapiro-WilkTest
       The Shapiro-Wilk W-Statistic test <http://en.wikipedia.org/wiki/Shapiro%E2%80%93Wilk_test> [Shapiro65] is
       considered to be among the most objective tests of normality [Royston92] and also one of the most
       powerful ones for detecting non-normality [Chen71].  Its statistic is essentially the roughly best
       unbiased estimator of population standard deviation to the sample variance [Dagostino71].  The test is
       mathematically complex and most implementations use several conventional approximations (as we do here),
       including Blom's formula for the expected value of the order statistics [Harter61] and transformation to
       standard normal distribution for evaluation, especially for large samples [Royston92].

               $pval = shapiro_wilk_test ([0.34, -0.2, 0.8, ...]);
               ($pval, $w_statistic) = shapiro_wilk_test ([0.34, -0.2, 0.8, ...]);

       This test may not be the best if there are many repeated values in the test distribution or when the
       number of points in the test distribution is very large, e.g. more than 5000.  The routine will carp
       about the latter, but not the former.  This particular implementation of the test also requires at least
       6 data points in the sample distribution and will croak otherwise.

   D'AgostinoK-SquaredTest
       The D'Agostino K-Squared test <http://en.wikipedia.org/wiki/D%27Agostino%27s_K-squared_test> is a good
       test against non-normality arising from kurtosis <http://en.wikipedia.org/wiki/Kurtosis> and/or skewness
       <http://en.wikipedia.org/wiki/Skewness> [Dagostino90].

               $pval = dagostino_k_square_test ([0.34, -0.2, ...]);
               ($pval, $ksq_statistic) = dagostino_k_square_test ([0.34, -0.2, ...]);

       The test statistic depends upon both the sample kurtosis and skewness, as well as the moments of these
       parameters from a normal population, as quantified by Pearson's coefficients [Pearson31].  These are
       transformed [Dagostino70,Anscombe83] to expressions that sum to the K-squared statistic, which is
       essentially chi-square-distributed with 2 degrees of freedom [Dagostino90].  The kurtosis transform, and
       thus the overall test, generally works best when the sample distribution has at least 20 data points
       [Anscombe83] and the routine will carp otherwise.

       Statistics::Normality - test whether an empirical distribution can be taken as being drawn from a
       normally-distributed population

       •   [Anscombe83]  Anscombe,  F.  J. and Glynn, W. J. (1983) DistributionoftheKurtosisStatisticB2forNormalSamples, Biometrika 70(1), 227-234.

       •   [Chen71] Chen, E. H. (1971) ThePoweroftheShapiro-WilkWTestforNormalityinSamplesfromContaminatedNormalDistributions, Journal of the American Statistical Association 66(336), 760-762.

       •   [Dagostino70]  D'Agostino,  R.  B. (1970) TransformationtoNormalityoftheNullDistributionofG1,
           Biometrika 57(3), 679-681.

       •   [Dagostino71] D'Agostino, R. B. (1971) AnOmnibusTestofNormalityforModerateandLargeSizeSamples, Biometrika 58(2), 341-348.

       •   [Dagostino90]  D'Agostino,  R. B. et al. (1990) ASuggestionforUsingPowerfulandInformativeTestsofNormality, American Statistician 44(4), 316-321.

       •   [Harter61] Harter, H. L. (1961) Expectedvaluesofnormalorderstatistics,  Biometrika  48(1/2),
           151-165.

       •   [Pearson31] Pearson, E. S. (1931) NotesonTestsforNormality, Biometrika 22(3/4), 423-424.

       •   [Royston92] Royston, J. P. (1992) ApproximatingtheShapiro-WilkW-testfornon-normality, Statistics
           and Computing 2(3) 117-119.

       •   [Shapiro65]  Shapiro,  S.  S.  and  Wilk,  M.  B. (1965) Ananalysisofvariancetestfornormality-completesamp1es, Biometrika 52(3/4), 591-611.

       You can find documentation for this module with the perldoc command.

           perldoc Statistics::Normality

       You can also look for information at:

       •   RT: CPAN's request tracker

           <http://rt.cpan.org/NoAuth/Bugs.html?Dist=Statistics-Normality>

       •   AnnoCPAN: Annotated CPAN documentation

           <http://annocpan.org/dist/Statistics-Normality>

       •   CPAN Ratings

           <http://cpanratings.perl.org/d/Statistics-Normality>

       •   Search CPAN

           <http://search.cpan.org/dist/Statistics-Normality/>

           use Statistics::Normality ':all';
           use Statistics::Normality 'shapiro_wilk_test';
           use Statistics::Normality 'dagostino_k_square_test';

       The subtleties and esoterica of various statistical tests for normality require some familiarity with the
       mathematical statistics literature.  We give rules-of-thumb for specific tests, where they exist, but it
       may be advisable to try several different tests to check the consistency of the conclusion.  It is
       probably also a good idea to check results graphically, either by direct plotting or by a Q-Q plot
       <http://en.wikipedia.org/wiki/Q-Q_plot>.  In general, small samples will often pass a normality test
       suggesting the possibility that there is insufficient information to detect departure from normal for
       such cases, should it exist.

       Each of the methods here is a frequentist test, i.e. one that tests against the null-hypothesis
       <http://en.wikipedia.org/wiki/Null_hypothesis> that the sample is normal.  In other words, a low p-value
       recommends rejecting the null.

       Version 0.01