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| In [[statistics]], the '''Shapiro–Wilk test''' tests the [[null hypothesis]] that a [[statistical sample|sample]] ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> came from a [[normal distribution|normally distributed]] population. It was published in 1965 by Samuel Shapiro and [[Martin Wilk]].<ref>{{cite journal
| | Greetings. Let me begin by telling you the author's title - Phebe. My family members lives in Minnesota and my family loves it. I used to be unemployed but now I am a librarian and the wage has been really fulfilling. To gather coins is what her family and her appreciate.<br><br>Feel free to surf to my website; [http://www.webmdbook.com/index.php?do=/profile-11685/info/ www.webmdbook.com] |
| |last=Shapiro |first=S. S.
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| |last2=Wilk |first2=M. B. |authorlink2=Martin Wilk
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| |year=1965
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| |title=An analysis of variance test for normality (complete samples)
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| |journal=[[Biometrika]]
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| |volume=52 |issue=3-4 |pages=591–611
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| |doi=10.1093/biomet/52.3-4.591 |jstor=2333709 | mr = 205384
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| }}</ref>
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| The [[test statistic]] is:
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| :<math>W = {\left(\sum_{i=1}^n a_i x_{(i)}\right)^2 \over \sum_{i=1}^n (x_i-\overline{x})^2}</math>
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| where
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| * <math>x_{(i)}</math> (with parentheses enclosing the subscript index ''i'') is the ''i''th [[order statistic]], i.e., the ''i''th-smallest number in the sample;
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| * <math>\overline{x} = \left( x_1 + \cdots + x_n \right) / n</math> is the sample mean;
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| * the constants <math>a_i</math> are given by<ref>{{cite journal
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| |last=Shapiro |first=S. S.
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| |last2=Wilk |first2=M. B. |authorlink2=Martin Wilk
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| |year=1965
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| |title=An analysis of variance test for normality (complete samples)
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| |journal=[[Biometrika]]
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| |volume=52 |issue=3-4 |pages=591–611
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| |doi=10.1093/biomet/52.3-4.591 |jstor=2333709 | mr = 205384
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| }} p. 593</ref>
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| ::<math>(a_1,\dots,a_n) = {m^\top V^{-1} \over (m^\top V^{-1}V^{-1}m)^{1/2}}</math>
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| :where
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| ::<math>m = (m_1,\dots,m_n)^\top\,</math>
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| :and <math>m_1,\ldots,m_n</math> are the [[expected value]]s of the [[order statistic]]s of [[independent and identically distributed random variables]] sampled from the standard normal distribution, and <math>V</math> is the [[covariance matrix]] of those order statistics.
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| The user may reject the null hypothesis if <math>W</math> is too small.<ref>{{cite journal
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| |last=Shapiro |first=S. S.
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| |last2=Wilk |first2=M. B. |authorlink2=Martin Wilk
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| |year=1965
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| |title=An analysis of variance test for normality (complete samples)
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| |journal=[[Biometrika]]
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| |volume=52 |issue=3-4 |pages=591–611
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| |doi=10.1093/biomet/52.3-4.591 |jstor=2333709 | mr = 205384
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| }} p. 605</ref>
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| It can be interpreted via a [[Q-Q plot]].
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| ==Interpretation==
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| Recalling that the null hypothesis is that the population is normally distributed, if the [[p-value]] is less than the chosen [[alpha level]], then the null hypothesis is rejected (i.e. one concludes the data are not from a normally distributed population). If the p-value is greater than the chosen alpha level, then one does not reject the null hypothesis that the data came from a normally distributed population. E.g. for an alpha level of 0.05, a data set with a p-value of 0.32 does not result in rejection of the hypothesis that the data are from a normally distributed population.<ref>{{cite web |url= http://www.jmp.com/support/faq/jmp2085.shtml |title=How do I interpret the Shapiro-Wilk test for normality?
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| |first= |last= |work=JMP |year=2004 |accessdate=March 24, 2012}}</ref>
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| ==See also==
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| * [[Anderson–Darling test]]
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| * [[Cramér–von Mises criterion]]
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| * [[Kolmogorov–Smirnov test]]
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| * [[Normal probability plot]]
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| * [[Ryan–Joiner test]]
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| * [[Watson test]]
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| {{primary sources|date=May 2012}}
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| ==References==
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| {{Ibid|date=May 2012}}
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| <references/>
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| ==External links==
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| * [http://www.answers.com/topic/samuel-sanford-shapiro Samuel Sanford Shapiro]
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| * [http://lib.stat.cmu.edu/apstat/R94 Algorithm AS R94 (Shapiro Wilk) FORTRAN code]
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| * [http://cran.us.r-project.org/doc/manuals/R-intro.html#Examining-the-distribution-of-a-set-of-data Shapiro–Wilk Normality Test in R]
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| {{Statistics}}
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| {{DEFAULTSORT:Shapiro-Wilk Test}}
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| [[Category:Normality tests]]
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Greetings. Let me begin by telling you the author's title - Phebe. My family members lives in Minnesota and my family loves it. I used to be unemployed but now I am a librarian and the wage has been really fulfilling. To gather coins is what her family and her appreciate.
Feel free to surf to my website; www.webmdbook.com