As a baseline, we’ll start with a top-down view of a normal distribution. The typical view is a side view. In the top-down view, the normal is the center of some concentric circles. In our graph, the concentric circles will have radiuses defined in terms of the statistical unit of measure, standard deviations. I’ve shown circles at 0,1,2, 3, and 5 standard distributions. The mean is shown at 0 standard distributions.

The core of the distribution is shown in orange. The horizontal view of the distribution defines the core as being between the inflection points (IP) of the normal curve. The core in a normal is the cylinder from the plane of the inflection points to the base of the distribution. The horizontal view can be rotated to align with the plane cutting the distribution for a particular dimension shown here as D1, D2, and D3. We only have three dimensions in this normal. With n dimensions, there would be n slices. With the normal, as long as the distribution is sliced through the mean, all the 2D projections would look the same. The normal is a symmetric distribution.

With a normal distribution, the mean, median, and mode have the same value. This and being symmetric is a property of the normal. More specifically, a non-skewed normal. A standard normal is not skewed.

The normal can be estimated with a Poisson distribution of at least twenty data points. The Poisson distribution will tend to the normal between twenty and thirty-six data points.

The normal is usually used in the snapshot dataset perspective, rather than in a time series sense. But, the time series sense is significant when you wonder if you’ve collected enough data. A dataset should be tested for normality. It is usually assumed. The tests for normality are weak.

Once the data achieve normality, the data tends to stay normal. The core and the outliers won’t move, and the standard deviation will stay the same. Until the data achieve normality, the distribution moves and resizes itself.

In the technology adoption lifecycle, the vertical phase will be the first time the normal is achieved. It will be a normal for the carried content. The horizontal, aka the early mainstreet market, has its own normal. The horizontal’s normal is for carrier components. The late market, likewise, has its own normal. In the late market, the focus is on carrier with the earlier carried discipline representing mass customization opportunities. The laggard and phobic phases are about form factors, carriers. Carried content may change is these phases. Carried content provides an opportunity to extend category life.

Preceding normality, the normal is skewed. In the next figure, I’ve put the skewed normal above the non-skewed normal.

Where the normal has a circular footprint, the skewed normal has an elliptical footprint. The median does not move. It tilts. This pushes the mode and the mean apart symmetrically around the median. The blue arrow shows how much the median tilts. The thick blue line shows the side view of the skewed normal. The core is shown in light orange. The tails are significant in the skewed normal. The skewed normal is asymmetrical. More on this later.

Each ellipse corresponds to the sigmas of our earlier diagram. But, the circular areas are the future. I’ve marked the outliers relative to the circular footprint of the non-skewed normal. The area I’m calling the deep outlier, the dark yellow population, is beyond what would be considered in the non-skewed normal. It would definitely be an error to collect data from that population, or since we sell to populations as we collect data from that population, it would be an error to sell to that segment of the population. Even after normality is achieved, outliers are more expensive than the revenues generated from that population.

The yellow populations are outliers, but they are outliers to the non-skewed normal. These outliers are shared by both distributions. The light green and even lighter green areas represent non-outlier populations that will be sold in the later normal, or as we sell to achieve normality. As the skewed normal achieves non-skewed normality, the ellipses will become circles. The edges located along the x-axis will move to the right. The tilted median will stand up vertically until it is perpendicular, and the mode and mean will converge to the median.

The ellipses would be thinner than shown. The probability mass under both distributions equals one so the ellipse would be less wide vertically than the circles. I had no idea about how wide those ellipses would be, but the figure is definitely wrong.

The skewed distribution exhibits kurtosis. I disagree with the idea that kurtosis has anything to do with peakedness. Other statisticians made this argument to me. The calculus view of the third moment disagrees as well. Kurtosis is about the tails and the shoulders as they relate to the cores. Some discussions ignore the shoulders. In this figure, I’ve included shoulders. I’ve used thick red lines and red text to highlight the components of the normal (N) and the skewed normal (SN). The normal only has one set of components. The skewed normal has two sets of components: one on the left, and another on the right.

I highlighted the shoulder of the normal. I highlighted the right and left shoulders of the skewed normal. And, lastly, I highlighted the right and left tails of the skewed normals.

The shoulders and tails are related to the cores. The normal core is a circle. The light orange ellipse of the skewed normal sits on top of it. I labeled both cores. The purple rectangle above the cores is the core of the skewed normal. The black core is the core of the non-skewed normal.

Kurtosis defines the curvature (κ) of the tails. I usually show these as circles defined as κ=1/r. These circles are tangents to the tails of the normal. In a normal, these circles are the same size on for both tails. In a skewed normal the circles are vastly different in size. These circles in both cases generate a topological object: A torus for the normal, and a ring cyclide for the skewed normal. These topological objects are generated as we rotate 360 degrees around the median or mean of the normal. I showed this topological object in dark orange. In this figure, I showed them as ellipses. The circular version made the diagram very large. The ellipse for the ring cyclide on the left side is large. On the right, it is very small. This is due to the horizontal slice through the 3D objects. The xy-plane used to produce the slice through both objects. Both objects are smooth and continuous so another slice through the median would show a smaller circle on the left and a larger circle on the right. At some rotational angle, both circles would be the same, as in both curvatures would be equal. The thick vertical line through the median turns out to be the slice in which both curvatures would be the same. This curvature would be the average curvature.

When I put the left portion of the torus in the figure, the blue line representing the side-view of the normal was incorrectly drawn. The peak should have been at the mode. This was the second surprise. The median has more frequency, but it is tilted at an angle, an angle that makes it less high than the mode. The mode being the highest was one of those not yet know pieces of knowledge.

I’ll attempt a multimodal normal with opposing long tails. I was going to try to illustrate a such a normal. There can be a multiplicity of centrality tuples, skews and long tails. With the tools I used now, that would be a challenge.

I’m looking at the Cauchy distribution now. There is no convergence. But, Cauchy sequences converge based on ε. You can pick your convergences. A footprint would be zeros. Different values of ε would different footprints, and different conclusions of the underlying logical argument in the triangle model sense of the width and depth of a conclusion.

The first thing that surprised me in this post was how a portion of the outliers, the deep outliers, of the skewed normal is too far away from my market. And, how other portions of the outliers are outliers in both distributions. Another example of writing to think, rather than writing to communicate. Sorry about that.

Care must be taken to ensure this if you are going to market to outliers. I won’t.

Enjoy.