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In statistics, an '''[[F-test|''F''-test]] for the [[null hypothesis]] that two [[normal distribution|normal]] populations have the same [[variance]]''' is sometimes used, although it needs to be used with caution as it can be sensitive to the assumption that the variables have this distribution.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


Notionally, any F-test can be regarded as a comparison of two variances, but the specific case being discussed in this article is that of two populations, where the [[test statistic]] used is the ratio of two [[sample variance]]s. This particular situation is of importance in [[mathematical statistics]] since it provides a basic exemplar case in which the [[F-distribution]] can be derived.<ref>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) ''Continuous Univariate Distributions, Volume 2'', Wiley. ISBN 0-471-58494-0 (Section 27.1)</ref>  For application in [[applied statistics]], there is concern{{Citation needed|date=May 2010}} that the test is so sensitive to the assumption of normality that it would be inadvisable to use it as a routine test for the equality of variances. In other words, this is a case where "approximate normality" (which in similar contexts would often be justified using the [[central limit theorem]]), is not good enough to make the test procedure approximately valid to an acceptable degree.
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== The test ==
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Let ''X''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub> and ''Y''<sub>1</sub>,&nbsp;...,&nbsp;''Y''<sub>''m''</sub> be [[independent and identically distributed]] samples from two populations which each have a [[normal distribution]]. The [[expected value]]s for the two populations can be different, and the hypothesis to be tested is that the variances are equal. Let
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


: <math> \overline{X} = \frac{1}{n}\sum_{i=1}^n X_i\text{ and }\overline{Y} = \frac{1}{m}\sum_{i=1}^m Y_i</math>
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


be the [[sample mean]]s.  Let
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


: <math> S_X^2 = \frac{1}{n-1}\sum_{i=1}^n \left(X_i - \overline{X}\right)^2\text{ and }S_Y^2 = \frac{1}{m-1}\sum_{i=1}^m \left(Y_i - \overline{Y}\right)^2 </math>
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


be the [[sample variance]]s.  Then the test statistic
==Demos==


: <math> F = \frac{S_X^2}{S_Y^2} </math>
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


has an [[F-distribution]] with ''n''&nbsp;&minus;&nbsp;1 and ''m''&nbsp;&minus;&nbsp;1 degrees of freedom if the [[null hypothesis]] of equality of variances is true.  Otherwise it has a non-central F-distribution.  The null hypothesis is rejected if ''F'' is either too large or too small.


== Properties==
* accessibility:
This F-test is known to be extremely sensitive to [[normal distribution|non-normality]],<ref>{{cite journal | last=Box | first=G.E.P. |authorlink=George E. P. Box| journal=Biometrika | year=1953 | title=Non-Normality and Tests on Variances  | pages=318&ndash;335 | volume=40 | jstor=2333350 | issue=3/4 | doi=10.1093/biomet/40.3-4.318}}</ref><ref>{{cite journal | last=Markowski | first=Carol A |author2=Markowski, Edward P. | year = 1990 | title=Conditions for the Effectiveness of a Preliminary Test of Variance | journal=The American Statistician | pages=322&ndash;326 | volume=44 | jstor=2684360 | doi=10.2307/2684360 | issue=4}}</ref> so [[Levene's test]], [[Bartlett's test]], or the [[Brown–Forsythe test]] are better tests for testing the equality of two variances. (However, all of these tests create experiment-wise [[type I error]] inflations when conducted as a test of the assumption of [[homoscedasticity]] prior to a test of effects.<ref>Sawilowsky, S. (2002). [http://digitalcommons.wayne.edu/coe_tbf/23 "Fermat, Schubert, Einstein, and Behrens–Fisher:The Probable Difference Between Two Means When σ<sub>1</sub><sup>2</sup> ≠ σ<sub>2</sub><sup>2</sup>"], ''Journal of Modern Applied Statistical Methods'', ''1''(2), 461&ndash;472.</ref>) F-tests for the equality of variances can be used in practice, with care, particularly where a quick check is required, and subject to associated diagnostic checking: practical text-books<ref>Rees, D.G. (2001) ''Essential Statistics (4th Edition)'', Chapman & Hall/CRC, ISBN 1-58488-007-4. Section 10.15</ref> suggest both graphical and formal checks of the assumption.
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[F-test]]s are used for other statistical [[hypothesis test|tests of hypotheses]], such as testing for differences in means in three or more groups, or in factorial layouts. These F-tests are generally not [[robust statistics|robust]] when there are violations of the assumption that each population follows the [[normal distribution]], particularly for small alpha levels and unbalanced layouts.<ref>Blair, R. C. (1981). "A reaction to ‘Consequences of failure to meet assumptions underlying the fixed effects analysis of variance and covariance.’" ''Review of Educational Research'', ''51'', 499–507.</ref> However, for large alpha levels (e.g., at least 0.05) and balanced layouts, the F-test is relatively robust, although (if the normality assumption does not hold) it suffers from a loss in comparative statistical power as compared with non-parametric counterparts.
==Test pages ==


==Generalization==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
The immediate generalization of the problem outlined above is to situations where there are more than two groups or populations, and the hypothesis is that all of the variances are equal. This is the problem treated by [[Hartley's test]] and [[Bartlett's test]].
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==See also==
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Goldfeld–Quandt test]]
*[[Url2Image|Url2Image (private Wikis only)]]
 
==Bug reporting==
==References==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .
{{reflist|30em}}
 
{{Statistics}}
 
{{DEFAULTSORT:F-Test Of Equality Of Variances}}
[[Category:Statistical ratios]]
[[Category:Statistical tests]]

Latest revision as of 23:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

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Bug reporting

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