What Is Leptokurtophobia?

And why does it matter?

Mon, 07/30/2012 - 13:17

Three years ago this month Quality Digest Daily published my column, “Do You Have Leptokurtophobia?” Based on the reaction to that column, it contained a message that was needed. In this column I would like to explain the symptoms of leptokurtophobia and the cure for this pandemic affliction.

Leptokurtosis is a Greek word that literally means “thin mound.” It was used to describe those probability models that have a central mound that is narrower than that of a normal distribution. In reality, due to the mathematics involved, a leptokurtic probability model is one that has heavier tails than the normal distribution. By a wide margin, most leptokurtic distributions are also skewed, and most skewed distributions will be leptokurtic.

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Comments

Submitted by leecqd on Wed, 08/01/2012 - 11:49

Re: Assumptions & data shaping - your article reminded me of a favorite quote from American geologist, T. C. Chamberlin:

“The fascinating impressiveness of rigorous mathematical analysis, with its atmosphere of precision and elegance, should not blind us to the defects of the premises that condition the whole process. There is, perhaps, no beguilement more insidious and dangerous than an elaborate and elegant mathematical process built upon unfortified premises”. (1899)

 

Such an elegant way to pop a balloon.

-lee c

Submitted by hweidig3 on Wed, 08/01/2012 - 13:28

Don:

The article states "Leptokurtosis ... means 'thin mound' ... used to describe those probability models that have a central mound that is narrower than that of a normal distribution."If the "thin mound" refers to the probablility graph, would it not be more precise to say it refers to "models that have a central mound that is wider or significantly less steep than that of a normal distribution"?  If I'm wrong, please explain.  Thx,

- HF

If I am dealing with a process that is in control, but whose data is nonnormal, my inclination is always to fit the underlying distribution and then set appropriate control limits. Then accurate process performance indices can be computed; StatGraphics and Minitab can in fact do this for nonnormal distributions. AIAG's SPC manual sanctions this approach for nonnormal systems.

Figures 1 and 2 represent two "processes" and therefore a bimodal distribution, for which no SPC scheme is appropriate. Figure 3 looks like it has numerous outliers in its upper tail, and therefore represents a system with assignable causes. If so, even if the underlying distribution were normal, it would not be possible to do SPC based on this process history. This underscores the fact that, as you said, the first step is to check for nonhomogeneity.

It might be interesting to model the situation in Figure 3 with a beta distribution (model for activity completion times in project management) as well as a lognormal distribution, but without the numerous outliers--if they can all be identified.

William, I do not know who taught you SPC, but you should ask for your money back since they did a particularly bad job. Virtually everything you say contains some sort of error. A process behavior chart is the universal technique for examining a data set for homogeneity. It does not impose any requirements upon the data. We do not have to fit a model to our data before we "do SPC." We do not need to verify the normality of the data prior to using a process behavior chart. We do not need to remove the outliers prior to placing the data on a chart. And as for fitting a lognormal, the uncertainty in the statistics means that you can fit a very wide spectrum of lognormals to these data, making a unique distribution impossible to justify.

I think the point is not that transformation is wrong per se, but that transformations are not necessary or even useful for industrial improvement studies.  Can data transformations help us understand something about the data before us?  Yes.  But what do they help us understand?  That is the real point of this discussion.  In Statistician lingo it’s about setting the correct frame.  A transformation of Non-Normal data can help us to calculate a Cpk/Ppk index.  But the real question is how useful is the index itself?  what can it tell us about the process that a simple time series plot can’t?  (answer:  nothing really, it just gives a precise number that quantifies one aspect of the process)  Process Capability indices yield no informative understanding to drive improvements.)

 

Transformations help us get mathematically precise answers to math questions when the data before us are Non Normal and/or non-homogenous; they do not help us get actionable answers to physics questions.  The industrial world is trying to solve physics problems not math problems.  It comes down to two things: 

­       - Real processes can be stable and capable even though they are non-homogenous and not Normally distributed. 

­       - Solving real industrial problems requires analytic studies not enumerative studies. 

 

Real world processes are often not well behaved like the processes used to teach mathematical statistics. A homogenous Normal distribution may be the “ideal” but it is not required to make quality product in a reliable and profitable manner.  Unfortunately it is required for many (distributional) statistical tests of significance.  Therefore we are taught that any distribution that is not Normal ‘must’ be transformed to use these common statistical tests.  This approach has transformed over the years into an incorrect belief that any process that is not homogenous and Normally distributed must be ‘bad’.  

 

Fortunately, analytic studies rely on replication and probability instead of distributional statistics. 

 

So the question isn’t whether or not one can transform the data or if there are situations where transforming the data might be helpful.  They are helpful in ENUMERATIVE studies and in limited ways in some complex modeling to assist in calculations.  The point of the article is to say that transformations are not helpful and often misleading in ANALYTIC studies.  Not transforming the data helps us see what is really going on.  I've solved hundreds of complex real world problems and never performed a transformation - they are not necessarry. 

 

A time series or other multi-variate plot is almost always the most valuable analysis tool the analytic study has…if you plot the hurricane data in time series you can clearly see the patterns – there are no ‘outliers’. There is the non homogenous effect of factors that cause hurricanes changing…transforming this data hides the causal system and suppresses knowledge.  This data can be found at:  http://www.aoml.noaa.gov/hrd/tcfaq/E11.html

Submitted by Dr Burns on Wed, 08/01/2012 - 19:00

Don's articles are always a great read.  However I suggest that the fear of leptokurtosis can be traced back to the surge of the Six Sigma scam in the 80's.  As with all scams, the driving force was money.  Six Sigma consultants and institutes made billions, while users suffered as a result of being conned by the nonsense. 

Despite Shewhart almost 100 years ago, having demonstrated that there was nothing to fear from non normal distributions, the masses swallowed the words of their heroes Smith and Harry.  Smith claimed the way to improve quality was to broaden the specification limits and Harry unbelievably "proved", based on stacks of discs, that every process drifts uncontrollably by +/- 1.5 sigma in the "long term" of 24 hours.  The vultures providing "quality" calculation software circled and fed on the ignorant masses' frenzy to click buttons to draw normal distributions over everything.  Millions of dollars were easy pickings, simply by conning gullible CEOs and Quality Managers that everything needed to be transformed to make it normal.

Greed has set quality backwards by 100 years.

I would agree that many of the six sigma trainers have perpetuated the use of transformations as they typically only teach what Dr. Deming called enumerative statistics.  All mathematics.  But I can't agree that they are the cause.  I have known too many statisticians whose passion is the math and if the distribution isn't Normal they can't deal with it unless they transfom the data.  This is typically becuase most of the commonly taught/used tests of significance are based on an assumption of Normality.  None of these statisticians have had more than a cursory 'knowledge' of six sigma and amost all were trained prior to Dr. Harry...it started a long time ago...

Yes, in 1982, page 132, "Out of the Crisis", Deming draws attention to the "deceptive and misleading" teaching of statistics as applied to production.  While at that time, Leptokurtophobia may have been a disease, Six Sigma was to turn it into a global pandemic.

Thanks for your comment. The word kurtosis was chosen about 100 years ago by an Englishman. He was trying to find a descriptive term for shape to go with skewness. He did not choose well. Kurtosis does not exactly describe either the mound or the tails of a distribution. One article that addressed this issue found that it best described the "absence of shoulders" for a distribution. So if you feel the word does not quite fit the job it has been given, you are in good company.

For those of  you unfamiliar with Don's books, you should try to get a copy of Understanding Statistical Process Control or USPC, pp 326-327. He shows two distributions, each of which has a skewness of 0.0000 and kurtosis of 2.0000. One looks like a house, the other like an elephant...His conclusion? "...we may properly conclude that the 'shape parameters' of skewness and kurtosis cannot even discriminate between an elephant and a house!"

And people think statisticians aren't funny! 

Submitted by bggraham on Fri, 08/17/2012 - 09:45

In the realm of using process behavior charts and assessing process capability, I stay clear of transforming data. Also, trying to find a best fit distribution model to a histogram that truly reflects more than one population present would be ‘silly’. That would be like sticking my head in a freezer and my feet in an oven at the same time and saying on average I feel great.

Over the years I’ve conducted many types of hypothesis tests ranging from simple to complex for designed experiments. In these cases, I’ve had good reason to ‘test for normality’ and at times to transform data. Every hypothesis test has a list of ‘underlying conditions’ which should be met in order to validate the analysis outcome. One of the common ‘requirements’ for using Parametric tests is that the populations being assessed can be modeled by the normal distribution; hence, the need to ‘test for normality’. If normality is ‘rejected’, then it may be appropriate to use an ‘equivalent’ Non-Parametric test depending upon several things; magnitude in departure from normality, sample size, and … . Non-Parametric tests tend to be ‘weaker’ than Parametric tests in detecting ‘significant differences’. In that case, it may be appropriate to use data transformation for all levels involved to validate the use of Parametric tests.

In summary, using the ‘right’ tool for the ‘right’ application at the ‘right’ time is key. Thanks again for the great article.

Regards, Bruce

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