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I am performing mvShapiro.Test from mvShapiroTest package. I get MVW = 0.9578, p = 0.0007656. I want to know whether the variable is normally distributed or not. One person in his thesis has showed that p-value below 0.05 do not violate normality. Is he correct? When p is below 0.05 in a univariate Shapiro test, that means it is not normal. Moreover, if it is non normal, what is best way to normalize it?

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    $\begingroup$ A comment below in reply to an answer implies that you have Likert scale data. That being so, you cannot even in principle achieve normal distributions! $\endgroup$ – Nick Cox Jul 1 '15 at 19:21
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    $\begingroup$ In addition to @NickCox's comments & the answer below, be aware that 'normalization' refers to transforming a variable to lie within [0,1], not with transforming it to be normally distributed. $\endgroup$ – gung Jul 1 '15 at 19:23
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A small p-value for a Shapiro-Wilk test indicates a departure from normality. Also, the "0.05" rule in hypothesis testing is just an arbitrary rule of thumb and should not be taken very seriously.

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  • $\begingroup$ So, what is the best way for me to normalize this variable (on Likert scale), please show simple code if normalized in R. $\endgroup$ – imran khan Jul 1 '15 at 19:19
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    $\begingroup$ +1 Agreed. Moreover, it is quite impossible to tell even from a very low P-value what is happening. It could even be that the distribution is roughly normal, but the sample size is large enough for slight non-normality to register. What is the purpose of the analysis? Marginal normality may not even be an assumption. What do the data look like? There are many, many threads here on testing normality and why it is less important than often supposed. $\endgroup$ – Nick Cox Jul 1 '15 at 19:20
  • $\begingroup$ Althoug, i am hopefully to perform PLS-SEM on my data, but i thought it is best to analyze multi variate normality also. The qq plot shows little deviated on both ends of the line. The study is causal study. While observing skewness and kurtosis, only on some items kurtosis is high more than 2 but below 4. $\endgroup$ – imran khan Jul 1 '15 at 19:22
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    $\begingroup$ As above, the normal is not a reference distribution for Likert scales. Even if some or all of your data were highly skewed, there is no useful transformation to consider. $\endgroup$ – Nick Cox Jul 1 '15 at 19:35
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    $\begingroup$ You don't need a reference for that. It's like asking if there's a paper that explains that sheep are not cows. $\endgroup$ – Nick Cox Jul 1 '15 at 20:55

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