Tests of normality - pros and cons of each, and what to do if they give different results?

BtG

New Member
#1
Hi everybody, sorry if this is a daft question but it's one I'm struggling to find an answer to (have googled etc. before posting here). I'm trying to up my stats knowledge to beyond "push this button - if-this-then-that", and I've been reading more about the various statistical tests for normality. I understand a bit about their limitations with large sample sizes, but the particular dataset I'm working with is small, so I would like to understand a bit more about how to interpret various figures.

My specific questions are:
a) what are the pros/cons of using z scores of skewness/kurtosis versus the Kolmogrov-Smirnov/Shapiro-Wilk tests to make a judgement about normality? As far as I can tell, they're both indications of the probability of a distribution being non-normal, but I'm not sure exactly how they work differently.
b) what does it mean/what should you do if they give different results (i.e. one suggests significant skew/kurtosis and the other doesn't? I get that it's useful to use one to confirm the results of the other, but what if the result is disconfirming?

To give an example for question b, I have a variable that returns the following result:

Skewness: .767
ZSkewness: 1.82
Kurtosis: -.364
ZKurtosis: 1.37
K-S value: .172 (p = .020)
S-W value: .910 (p = .013)

So as far as I understand it, the z scores suggest that the variable isn't significantly non-normal (because the z values are less than 1.96), but the K-S/S-W tests suggest that it is significantly non-normal? What does this pattern of values mean, and which is better to go with? Have attached an image of the histogram & descriptives if that helps to make sense of it......

Many thanks for your help.
 

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Karabiner

TS Contributor
#2
what are the pros/cons of using z scores of skewness/kurtosis versus the Kolmogrov-Smirnov/Shapiro-Wilk tests to make a judgement about normality? As far as I can tell, they're both indications of the probability of a distribution being non-normal, but I'm not sure exactly how they work differently.
Statistical tests about normality have the Null hypothesis that sample data are fom a normally
distributed population. Therefore, if sample is increased, at some point you will necessarily
achive a significant resuit (this assumes that ususally variables are not exactely distributed normally
in the population, see for example here) . A non-significant result will therefore be
a type 2 errror. So it is an intersting question why we should bother about normality tests.
They do not tell us whether the non-npormality in the population is maybe important or not.

So why should we be interested in normality at all? Normality of variables or of unconditional
measurements is not of interest for nearly any statisticial procedure. There are some situations in
which we would like to know about normality of variables within subgroups (t-test, for example),
or about the normality of model residuals (linear regression, anova for example), but only if samples
size is very small. Unfortunately, as mentioned above, small sample sizes will produce non-significant
normality tests, even if deviations could be relevant.

With kind regards

Karabiner
 
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BtG

New Member
#3
Thanks for your reply. I totally get that there are lots of reasons not to be interested in normality. But I guess just for technical knowledge I'd like to better understand the pros/cons of each test & what to do if they return different results.
 

Miner

TS Contributor
#4
There is a good explanation here. The short answer for different results is the difference in power as well as that each is more sensitive to different portions of the distribution. This holds true for other tests such as the Anderson-Darling test, which is more sensitive to the tail areas.