Range of values of skewness and kurtosis for normal distribution

Question

I want to know that what is the range of the values of skewness and kurtosis for which the data is considered to be normally distributed.

I have read many arguments and mostly I got mixed up answers. Some says for skewness $(-1,1)$ and $(-2,2)$ for kurtosis is an acceptable range for being normally distributed. Some says $(-1.96,1.96)$ for skewness is an acceptable range. I found a detailed discussion here: What is the acceptable range of skewness and kurtosis for normal distribution of data regarding this issue. But I couldn't find any decisive statement.

What is the basis for deciding such an interval? Is this a subjective choice? Or is there any mathematical explanation behind these intervals?

Answer 1

The original post misses a couple major points: (1) No "data" can ever be normally distributed. Data are necessarily discrete. The valid question is, "is the process that produced the data a normally distributed process?" But (2) the answer to the second question is always "no", regardless of what any statistical test or other assessment based on data gives you. Normally distributed processes produce data with infinite continuity, perfect symmetry, and precisely specified probabilities within standard deviation ranges (eg 68-95-99.7), none of which are ever precisely true for processes that give rise to the data that we can measure with whatever measurement device we humans can use.

So you can never consider data to be normally distributed, and you can never consider the process that produced the data to be a precisely normally distributed process. But, as Glen_b indicated, it might not matter too much, depending on what it is that you are trying to do with the data.

Skewness and kurtosis statistics can help you assess certain kinds of deviations from normality of your data-generating process. They are highly variable statistics, though. The standard errors given above are not useful because they are only valid under normality, which means they are only useful as a test for normality, an essentially useless exercise. It would be better to use the bootstrap to find se's, although large samples would be needed to get accurate se's.

Also, kurtosis is very easy to interpret, contrary to the above post. It is the average (or expected value) of the Z values, each taken to the fourth power. Large |Z| values are outliers and contribute heavily to kurtosis. Small |Z| values, where the "peak" of the distribution is, give Z^4 values that are tiny and contribute essentially nothing to kurtosis. I proved in my article https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4321753/ that kurtosis is very well approximated by the average of the Z^4 *I(|Z|>1) values. Hence kurtosis measures the propensity of the data-generating process to produce outliers.

Answer 2

What you seem to be asking for here is a standard error for the skewness and kurtosis of a sample drawn from a normal population. Note that there are various ways of estimating things like skewness or fat-tailedness (kurtosis), which will obviously affect what the standard error will be. The most common measures that people think of are more technically known as the 3rd and 4th standardized moments.

It is worth considering some of the complexities of these metrics. The typical skewness statistic is not quite a measure of symmetry in the way people suspect (cf, here). The kurtosis can be even more convoluted. It has a possible range from $[1, \infty)$, where the normal distribution has a kurtosis of $3$. As a result, people usually use the "excess kurtosis", which is the ${\rm kurtosis} - 3$. Then the range is $[-2, \infty)$. However, in practice the kurtosis is bounded from below by ${\rm skewness}^2 + 1$, and from above by a function of your sample size (approximately $24/N$). In addition, the kurtosis is harder to interpret when the skewness is not $0$. These facts make it harder to use than people expect.

For what it's worth, the standard errors are:

\begin{align} SE({\rm skewness}) &= \sqrt{\frac{6N(N-1)}{(N-2)(N+1)(N+3)}} \\[10pt] SE({\rm kurtosis}) &= 2\times SE({\rm skewness})\sqrt{\frac{N^2-1}{(N-3)(N+5)}} \end{align}

Setting aside the issue of whether we can differentiate the skewness and kurtosis of our sample from what would be expected from a normal population, you can also ask how big the deviation from $0$ is. The rules of thumb that I've heard (for what they're worth) are generally:

• $<|.5|$ small
• $[|.5|, |1|)$ medium
• $\ge |1|$ large

A good introductory overview of skewness and kurtosis can be found here.

Answer 3

[In what follows I am assuming you're proposing something like "check sample skewness and kurtosis, if they're both within some pre-specified ranges use some normal theory procedure, otherwise use something else".]

There's a host of aspects to this, of which we'll only have space for a handful of considerations. I'll begin by listing what I think the important issues may be to look at before leaping into using a criterion like this. I will attempt to come back and write a little about each item later:

Issues to consider

1. How badly would various kinds of non-normality matter to whatever we're doing?

2. How hard is it to pick up those deviations using ranges on sample skewness and kurtosis?

   One thing that I agree with in the proposal - it looks at a pair of measures related to effect size (how much deviation from normality) rather than significance. In that sense it will come closer to addressing something useful that a formal hypothesis test would, which will tend to reject even trivial deviations at large sample sizes, while offering the false consolation of non-rejection of much larger (and more impactful) deviations at small sample sizes. (Hypothesis tests address the wrong question here.)

   Of course at small sample sizes it's still problematic in the sense that the measures are very "noisy", so we can still be led astray there (a confidence interval will help us see how bad it might actually be).

   It doesn't tell us how a deviation in skewness or kurtosis relates to problems with whatever we want normality for -- and different procedures can be quite different in their responses to non-normality.

   It doesn't help us if our deviation from normality is of a kind to which skewness and kurtosis will be blind.

3. If you're using these sample statistics as a basis for deciding between two procedures, what is the impact on the properties of the resulting inference (e.g. for a hypothesis test, what do your significance level and power look like doing this?)

4. There are an infinite number of distributions that have exactly the same skewness and kurtosis as the normal distribution but are distinctly non-normal. They don't even need to be symmetric! How does the existence of such things impact the use of such procedures? Is the enterprise doomed from the start?

5. How much variation in sample skewness and kurtosis could you see in samples drawn from normal distributions? (What proportion of normal samples would we end up tossing out by some rule?)

   [In part this issue is related to some of what gung discusses in his answer.]

6. Might there be something better to do instead?

Finally, if after considering all these issues we decide that we should go ahead and use this approach, we arrive at considerations deriving from your question:

7. what are good bounds to place on skewness and on kurtosis for various procedures? What variables do we need to worry about in which procedures?

   (e.g. if we're doing regression, note that it's incorrect to deal with any IV and even the raw DV this way -- none of these are assumed to have been drawn from a common normal distribution)

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I will come back and add some thoughts, but any comments / questions you have in the meantime might be useful.