More On Skew and Kurtosis

After the last blog post, Donuts, I was still puzzled about where skew and kurtosis come from. I’ve chased enough rabbits into their holes with this one. I’m tired of the obsession, so I’ll write this one up and let go of it until I cross paths with again down the road.

It was stats. It became math. It’s on its way to becoming a set of tools like the black swan that can be applied within product management in the sense that here is the investment. How do we code down the kurtosis? I found mentions of skew risk and kurtosis risk. They are not a playground yet.

Skew and kurtosis were and still are descriptive. Later came the “summary statistics” that our spreadsheets generate for us, but read that again “summary statistics.” With kurtosis one number is describing two things. Well, those two things are connected or are part of one thing, a thing never described anywhere in the literature I’ve cruised through, another donut. Then, there is the matter of that angle, that I found a hint for after I found it own my own. The angle accounts for the two kurtoses, and the new donut.

The literature talks about moments. Skew is the third moment; kurtosis, the forth. Then, there is another view that talks integrals, and another that talks derivatives.

For myself, it boils down to derivatives being about inflection points. Three of them: 1) a global maxima,  and 2) two concavity change inflection points. That’s all there is for all of that calculus. There are a few more concavity changes, but  no more points. The fifth and sixth derivatives sit on top of the third and forth. The drive the curve, but don’t present us with any additional inflection points.

Inflection Points on the Standard Normal

All the mentions of leptokurtic, platykurtic, and mesocratic  are just terminology from long ago lacking in any numeric definition or reality. Some times we are told the data has these characteristics, but we need to keep in mind that we are describing a curve, rather than the data. We use summary statistics and distributions to make the data itself disappear. So whatever is going on is not the result of the data doing anything. The data stands around in lines, we call histograms.

On of my early pursuits was a search for slant asymptotes. Well, there are none. There is a horizontal asymptote. It is a cubic rather than a straight line. The cubic crosses the x-axis at the origin. It leaves us wondering where our convergences are with the line “formerly known as the x-axis.” Anyway, when you have a horizontal asymptote, you won’t have any slant asymptotes.

Horizontal Asympote

Next I looked to extrapolate something I read about setting up bins in regard to a given range of numbers. The binomial approximates the normal when the bins capture the data evenly.

Decision Tree - All bins

The bin widths had to be the same even if the data width doesn’t completely fill those bins. Maybe we only have data to fill the right half of the base of the decision tree.

Decision Tree - Some bins

I didn’t draw a distribution for this decision tree. The distribution will be skewed with a long tail to the left and the short tail to the right. The first box plot below shows what the distribution resulting from the above decision tree will look like. The second box plot is not skewed and is shown for comparison purposes only.


When looking at box plots, if the line dividing the box does not divide the box into two equal size partitions, the distribution is skewed. Likewise, if the tails are not of equal length, even if the box has equal partitions, the distribution is skewed. Likewise if the outliers, not show, are not of equal distance from the mode, the distribution is skewed. These outlier skews are sensitive. Measures of coskewness and cokurtosis are about sensitivity in the financial/investment domain. Beware of outliers. I’ve said it before, say no to sales when they present you with deal demands from outliers.

The boxplot view gives a hint to the angle driving skew and kurtosis. Keep in mind that without skew, there is no kurtosis, or the kurtosis has a summary statistic value of 3, aka no kurtosis.

The Angle

I ran some lines out from the unskewed mode and skewed mode. The angle between them ties to kurtosis. I didn’t read this anywhere, but later did find some diagrammatic hints from other writers out on the internet. Notice that the mean never moves and that the vertical line labelled mode is also the mean in the unskewed case. Notice that there are two different kurtosis measures apparent in this view. This is where the summary statistic goes off in the weeds unless it is an index to both kurtoses. Given that we started with the standard normal and deformed it in a consistent manner, the two kurtoses should be correlated and indexed. I’ve not come across such.

Kurtoses are measure by curvature. The Kurtosis  curves are intrinsic curves. There is no controls off the line as in the Bezier curves we’ve discussed in the past. Curvature is a circle generated with radi of the recipricol of radius, aka 1/r.

The Curvatures

Notice the gap between the blue line and the red one. I  couldn’t make that circle big enough. But, this two dimensional view misses that there are kurtoses in every direction around the distribution. Here we’ve show the largest and the smallest. Those encompassing the distribution would be smaller than the largest and larger then the smallest. Sweeping these kurtoses around would give us a lopsided donut.


I leave it up to your imagination to sweep the ellipse around the core of the distribution to form the donut. I made a mistake by limiting the redlines to the tails of the distribution indicated by the outer circle. The actual radi would extend beyond the circle for the longer tails and not touch the outer circle for the shorter tails.

Have fun with it.






6 Responses to “More On Skew and Kurtosis”

  1. Convergence and Divergence—Adoption, De-​adoption, Readoption | Product Strategist Says:

    […] side or another. When it leans, the mean stays put, but the mode moves by some angle theta. My last blog post on kurtosis mentions theta relative to one of the […]

  2. The Tracy-Windom Distribution and the Technology Adoption Lifecycle Phases. | Product Strategist Says:

    […] more on leaning and theta, see Convergence and Divergence—Adoption, De- adoption, Readoption, and More On Skew and Kurtosis. They are taking the Tracy-Widom distribution as a given here, rather than a transformation of the […]

  3. Kurtosis Risk | Product Strategist Says:

    […] that, and getting too far away from what I was writing about in those posts. mc spacer retweeted More On Skew and Kurtosis. I reread the post and decided to conquer kurtosis risk. The exploration was […]

  4. The Curvature Donut | Product Strategist Says:

    […] Risk, we looked at how skew is a tilt of the mean by some angle theta. This tilt is much clearer in More On Skew and Kurtosis. That skewness moves the peak and the inflection points but the curve stays […]

  5. A Quick Viz, Long Days | Product Strategist Says:

    […] The tiny thumbnail in the middle of the thumbnails at the bottom right of the figure shows a negatively skewed normal, but in another interpretation, the distribution is four separate normals. Where I mentioned theta, the associated angle quantifies the kurtosis.  […]

  6. Kurtosis, Another Definition | Product Strategist Says:

    […] curve, which I’ve not used since I found kurtosis to be the curvature of the tails. See More On Skew and Kurtosis. I’m still lost as to how the kurtosis statistic translates into the curvatures of skewed […]

Leave a Reply

Fill in your details below or click an icon to log in: Logo

You are commenting using your account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )


Connecting to %s

%d bloggers like this: