Should we teach kurtosis in an applied statistics course? If so, how?

Question

Central tendency, spread and skewness can all be defined relatively well, at least on an intuitive basis; the standard mathematical measures of these things also correspond relatively well to our intuitive notions. But kurtosis seems to be different. It's very confusing and it doesn't match well with any intuition about distributional shape.

A typical explanation of kurtosis in an applied setting would be this extract from Applied statistics for business and management using Microsoft Excel $^{[1]}$:

Kurtosis refers to how peaked a distribution is or conversely how flat it is. If there are more data values in the tails, than what you expect from a normal distribution, the kurtosis is positive. Conversely if there are less data values in the tails, than you would expect in a normal distribution, the kurtosis is negative. Excel cannot calculate this statistic unless you have at least four data values.

Aside from the confusion between "kurtosis" and "excess kurtosis" (as in this book, it is common to use the former word to refer to what others author call the latter), the interpretation in terms of "peakedness" or "flatness" is then muddled by the switch of attention to how many items of data are in the tails. Considering both "peak" and "tails" is necessary — Kaplansky$^{[2]}$ complained in 1945 that many textbooks of the time erroneously stated kurtosis was to do with how high the peak of the distribution is compared to that of a normal distribution, without considering the tails. But clearly having to consider the shape both at the peak and in the tails makes the intuition harder to grasp, a point the extract quoted above skips over by seguing from peakedness to heaviness of tails as if these concepts were the same.

Moreover this classical "peak and tails" explanation of kurtosis only works well for symmetric and unimodal distributions (indeed, the illustrated examples in that text are all symmetric). Yet the "correct" general way to interpret kurtosis, whether in terms of "peaks", "tails" or "shoulders", has been disputed for decades.$^{[2][3][4][5][6]}$

Is there an intuitive way of teaching kurtosis in an applied setting which will not hit contradictions or counterexamples when a more rigorous approach is taken? Is kurtosis even a useful concept at all in the context of these kind of applied data analysis courses, as opposed to in mathematical statistics classes? If "peakedness" of a distribution is an intuitively useful concept, should we teach it by way of L-moments$^{[7]}$ instead?

$[1]$ Herkenhoff, L. and Fogli, J. (2013). Applied statistics for business and management using Microsoft Excel. New York, NY: Springer.

$[2]$ Kaplansky, I. (1945). "A common error concerning kurtosis". Journal of the American Statistical Association, 40(230): 259.

$[3]$ Darlington, Richard B (1970). "Is Kurtosis Really 'Peakedness'?". The American Statistician 24(2): 19–22

$[4]$ Moors, JJA. (1986) "The meaning of kurtosis: Darlington reexamined". The American Statistician 40(4): 283–284

$[5]$ Balanda, Kevin P. and MacGillivray, H.L. (1988). "Kurtosis: A Critical Review". The American Statistician 42(2): 111–119

$[6]$ DeCarlo, L. T. (1997). "On the meaning and use of kurtosis". Psychological methods, 2(3), 292. Chicago

$[7]$ Hosking, J.R.M. (1992). "Moments or L moments? An example comparing two measures of distributional shape". The American Statistician 46(3): 186–189

Answer 1

Kurtosis is really pretty simple ... and useful. It is simply a measure of outliers, or tails. It has nothing to do with the peak whatsoever - that definition must be abandoned.

Here is a data set:
0, 3, 4, 1, 2, 3, 0, 2, 1, 3, 2, 0, 2, 2, 3, 2, 5, 2, 3, 999

Notice that '999' is an outlier.

Here are the $z^4$ values from the data set:

0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00,0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 0.00, 360.98

Notice that only the outlier gives a $z^4$ that is noticeably different from 0.

The average of these $z^4$ values is the kurtosis of the empirical distribution (subtract 3 if you like, it doesn't matter for the point I am making): 18.05

It should be obvious from this calculation that the data near the "peak" (the non-outlier data) contribute almost nothing to the kurtosis statistic.

Kurtosis is useful as a measure of outliers. Outliers are important to elementary students and therefore kurtosis should be taught. But kurtosis has virtually nothing to do with the peak, whether it is pointy, flat, bimodal or infinite. You can have all the above with small kurtosis and all of the above with large kurtosis. So it should NEVER be presented as having anything to do with the peak, because that will be teaching incorrect information. It also makes the material needless confusing, and seemingly less useful.

Summary:

1. kurtosis is useful as a measures of tails (outliers).
2. kurtosis has nothing to do with the peak.
3. kurtosis is practically useful and should be taught, but only as a measure of outliers. Do not mention peak when teaching kurtosis.

This article explains clearly why the "Peakedness" definition is now officially dead.

Westfall, P.H. (2014). "Kurtosis as Peakedness, 1905 – 2014. R.I.P." The American Statistician, 68(3), 191–195.

Answer 2

While the question is somewhat vague, it is interesting. At what levels is kurtosis taught? I remember it being mentioned in a (master's level) course in linear models (long time ago, based on first edition of Seber's book). It was not an important topic, but it enters in topics like studying the (lack of) robustness of the Likelihood ratio test (F-test) of equality of variances, where (from memory) correct level asymptotically depends on having same kurtosis as the normal distribution, which is too much to assume! We saw a paper (but I never read it with details) http://www.jstor.org/stable/4615828?seq=1#page_scan_tab_contents by Oja, which tries to find out what skewness, kurtosis and such really measures.

Why do I find this interesting? Because I have been teaching in latin america, where it seems that skewness & kurtosis are taught by many as important topics, and trying to tell post-graduate students (many from economy) that kurtosis is a bad measure of form of a distribution (mainly because sampling variability of fourth powers simply is to large), was difficult. I was trying getting them to use QQplots instead. So, to some of the commenters, yes, this is taught someplaces, probably to much!

By the way, this is not only my opinion. The following blog post https://www.spcforexcel.com/knowledge/basic-statistics/are-skewness-and-kurtosis-useful-statistics contains this citation (attributed to Dr. Wheeler):

In short, skewness and kurtosis are practically worthless. Shewhart made this observation in his first book. The statistics for skewness and kurtosis simply do not provide any useful information beyond that already given by the measures of location and dispersion.

We should teach better techniques to study forms of distributions! such as QQplots (or relative distribution plots). And, if somebody still needs numerical measures, measures based on L-moments are better. I will quote one passage from the paper J R Statist Soc B (1990) 52, No 1, pp 105--124 by J R M Hosking: "L-moments: Analysis and Estimation of Distribution using Linear Combination of Order Statistics", page 109:

An alternative justification of these interpretations of L-moments may be based on the work of Oja (1981), Oja defined intuitively reasonable criteria for one probability distribution on the real line to be located further to the right (more dispersed, more skew, more kurtotic) than another. A real-valued functional of a distribution that preserves the partial ordering of distributions implied by these criteria may then reasonably be called a 'measure of location (dispersion, skewness, kurtosis)'. It follows immediately from Oja's work that $\lambda_1$ and $\lambda_2$ , in Oja's notation, $\mu(F)$ and $\frac12 \sigma_1(F)$, are measures of location and scale respectively. Hosking (1989) shows that $\tau_3$ and $\tau_4$ are, by Oja's criteria, measures of skewness and kurtosis respectively.

(For the moment, I refer to the paper for the definitions of these measures, they are all based on L-moments.) The interesting thing is that, the traditional measure of kurtosis, based on fourth moments, is not a measure of kurtosis in the sense of Oja! (I will edit in references for that claim when I can find it).

Answer 3

I my opinion, the skewness coefficient is useful to motivate the terms: positively skewed and negatively skewed. But, that is where it stops, if your goal is to assess normality. Classical measures of skewness and kurtosis often fail to capture various types of deviation away from normality. I usually advocate to my students to use graphical techniques to assess it is reasonable to assess normality, such as a qq-plot or a normal probability plot. Also with an adequately sized sample, a histogram can also be used. Boxplots are also useful to identify outliers or even heavy tails.

This is inline with the recommendations a 1999 task force of the APA:

"Assumptions. You should take efforts to assure that the underlying assumptions required for the analysis are reasonable given the data. Examine residuals carefully. Do not use distributional tests and statistical indexes of shape (e.g., skewness, kurtosis) as a substitute for examining your residuals 'graphically. Using a statistical test to diagnose problems in model fitting has several shortcomings. First, diagnostic significance tests based on summary statistics (such as tests for homogeneity of variance) are often impractically sensitive; our statistical tests of models are often more robust than our statistical tests of assumptions. Second, statistics such as skewness and kurtosis often fail to detect distributional irregularities in the residuals. Third, statistical tests depend on sample size, and as sample size increases, the tests often will reject innocuous assumptions. In general, there is no substitute for graphical analysis of assumptions."

Reference: Wilkinson, L., & Task Force on Statistical Inference. (1999). Statistical methods in psychology journals: Guidelines and explanations. American Psychologist, 54, 594-604.

Answer 4

Depending on how applied the course is, the question of accuracy of estimates might come up. The accuracy of the variance estimate depends strongly on kurtosis. The reason this happens is that with high kurtosis, the distribution allows rare, extreme potentially observable data. Thus the data-generating process will produce very extreme values in some samples, and not so extreme values in others. In the former case, you get a very large variance estimate, and in the latter, a small variance estimate.

If the outdated and incorrect "peakedness" interpretation were eliminated, and the focus given entirely to outliers (i.e., rare, extreme observables) instead, then it would be easier to teach kurtosis in introductory courses. But people twist themselves into knots trying to justify "peakedness" because it is (incorrectly) stated that way in their textbooks, and they miss the real applications of kurtosis. These applications mostly relate to outliers, and of course outliers are important in applied statistics courses.

Answer 5

Frankly, I don't understand why people want to complicate simple things. Why not just show the definition (stolen from Wikipedia): $$\operatorname{Kurt}[X] = \operatorname{E}\left[\left(\frac{X - \mu}{\sigma}\right)^4\right] = \frac{\mu_4}{\sigma^4} = \frac{\operatorname{E}[(X-\mu)^4]}{(\operatorname{E}[(X-\mu)^2])^2}, $$

You can replace the expectation operator with sum based estimators $\frac 1 n \sum_{i=1}^n$, of course. It helps to discuss the units of measure of $\mu,\sigma^2,\mu_4$, and show why the fourth moment should be scaled by the square of the variance to make kurtosis the dimensionless measure, i.e. a shape parameter. So, we have now location $\mu$, scale $\sigma^2$ and any number of parameters to describe the shape such as skew and kurtosis. I'd always start with equations. Supposedly easy to understand explanations in plain English only make everything more confusing. Verbosity $\ne$ clarity.