I am trying to determine if my data is normal. I am using R.

I run the jarque bera test that has a NULL hypothesis of Normality

    Jarque Bera Test

data:  dat
  X-squared = 4.6747, df = 2, p-value = 0.09658

since the pvalue is > .05 I cannot reject he null so the data IS normal

I also run the shapiro wilk test with a null of NORMALITY

shapiro.test(dat) #Ho is normal

Shapiro-Wilk normality test

data:  dat
W = 0.9149, p-value = 0.001375

so I reject the Null and the data is not NORMAL

THEN when I look at a QQnorm I seeenter image description here

and that does not look normal.

So 2 of 3 tests say the data is not normal is that enough to say it is not normal? What do you think?

  • 2
    $\begingroup$ Why are you testing normality? $\endgroup$
    – Glen_b
    Mar 28, 2015 at 2:09
  • $\begingroup$ Even if we have to repeat it infinitely: a high p value does not imply a true null. So you will never be able to show normality using these tests. $\endgroup$
    – Michael M
    Mar 28, 2015 at 9:50
  • $\begingroup$ hi Glen and Michael - I am testing for normality because I wan to see if I can use a t test to test if the differences between means and normality is required. MIchael - I am not sure what you mean. The p value is lowest level of significance at which you can accept the null so if pvalue is .00001 and i use a .05 significnace then I cannot accept null. DO you agree? $\endgroup$
    – joesyc
    Mar 29, 2015 at 17:01

1 Answer 1


So your first test, the Jarque-Bera test doesn't test for normality. It tests if the skewness and kurtosis is the same as that of a normal distribution. That is not the same.

The Shapiro-Wilk test does look at normality.

Looking at your data, it does not look normal.

First your qq-plot deviates quite a bit from the line. Moreover, going by the Shapiro-Wilk test -- it's not even close -- your p-value is $\approx .001$. That's very significant. If it was 0.04, that would be more of a subjective call...

  • $\begingroup$ wikipeida does argue: "If the data come from a normal distribution, the JB statistic asymptotically has a chi-squared distribution with two degrees of freedom, so the statistic can be used to test the hypothesis that the data are from a normal distribution." $\endgroup$
    – charles
    Mar 28, 2015 at 6:26
  • 1
    $\begingroup$ @charles -- looking at the statistic, it seems that it particularly looks at skewness and kurtosis. The null is a joint hypothesis that the skewness and excess kurtosis is zero. What this means is that if you fail to reject the null, your skewness and kurtosis don't necessarily violate what you would find under a normal distribution. However, if you fail to reject the null, all that means is that your skewness and kurtosis are aligned with what you would get under normality. For shapiro-wilks, it means your actual observations are aligned with what you would get under normality. $\endgroup$ Mar 28, 2015 at 6:53
  • $\begingroup$ I upvoted the answer. And seems reasonble. But wanted to emphasize that there are resources out there suggesting it can be used for normality. $\endgroup$
    – charles
    Mar 28, 2015 at 9:16
  • $\begingroup$ NIck/Charles - so what does this mean for the data. IS it normal or not normal? $\endgroup$
    – joesyc
    Mar 29, 2015 at 17:03

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