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I'm examining a part of my dataset containing 46840 double values ranging from 1 to 1690 grouped in two groups. In order to analyze the differences between these groups I started by examining the distribution of the values in order to pick the right test.

Following a guide on testing for normality, I did a qqplot, histogram & boxplot.

enter image description here

enter image description here enter image description here

This doesn't seem to be a normal distribution. Since the guide states somewhat correctly that a purely graphical examination isn't sufficient I also want to test the distribution for normality.

Considering the size of the dataset and the limitation of the shapiro-wilks test in R , how should the given distribution be tested for normality and considering the size of the dataset, is this even reliable? (See accepted answer to this question)

Edit:

The limitation of the Shapiro-Wilk test I'm referring to is that the dataset to be tested is limited to 5000 points. To cite another good answer concerning this topic:

An additional issue with the Shapiro-Wilk's test is that when you feed it more data, the chances of the null hypothesis being rejected becomes larger. So what happens is that for large amounts of data even very small deviations from normality can be detected, leading to rejection of the null hypothesis event hough for practical purposes the data is more than normal enough.

[...] Luckily shapiro.test protects the user from the above described effect by limiting the data size to 5000.

As to why I am testing for normal distribution in the first place:

Some hypothesis tests assume normal distribution of the data. I want to know whether or not I can use these tests.

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There's no point testing; every test of any use at all, a any reasonable significance level will clearly reject. Whatever guide you're reading has misled you. What do you mean by 'reliable' exactly. What 'limitation' of the Shapiro-Wilk do you refer to? I would very nearly agree with the statement in the answer you link to ... "I have never come across a situation where a normal test is the right thing to do" (I've seen at least once situation where I think it's the right thing to do, but people almost always do it for bad reasons). –  Glen_b Apr 3 at 11:04
    
Answered some of your questions in an edit –  Rickyfox Apr 3 at 11:15
    
@Glen_b: incidentally, I found myself using Shapiro–Wilk the other day to quantify the evidence against the null, which someone over on Academia had mistakenly assumed to be greater than it was from a sample of grades. I wonder if that was defensible usage. –  Nick Stauner Apr 3 at 11:22
    
@NickStauner my response grew much too long for a single comment and I don't want to hijack this question with a string of comments about your post there. Possibilities: We talk in chat, or you post a question about it (to which I could post an extensive answer), or we discuss it in some other way, such as email. –  Glen_b Apr 3 at 21:31
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1 Answer 1

up vote 10 down vote accepted

I don't see why you'd bother. It's plainly not normal – in this case, graphical examination appears sufficient to me. You've got plenty of observations from what appears to be a nice clean gamma distribution. Just go with that. it if you must – I'll recommend a reference distribution.

x=rgamma(46840,2.13,.0085);qqnorm(x);qqline(x,col='red')
enter image description here

hist(rgamma(46840,2.13,.0085))

boxplot(rgamma(46840,2.13,.0085))

As I always say, "See Is normality testing 'essentially useless'?," particularly @MånsT's answer, which points out that different analyses have different sensitivities to different violations of normality assumptions. If your distribution is as close to mine as it looks, you've probably got skew $\approx1.4$ and kurtosis $\approx5.9$ ("excess kurtosis" $\approx2.9$). That's liable to be a problem for a lot of tests. If you can't just find a test with more appropriate parametric assumptions or none at all, maybe you could transform your data, or at least conduct a sensitivity analysis of whatever analysis you have in mind.

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