## The Curvature Donut

In last month’s The Cones of Normal Cores, I was visualizing the cones from the curvatures of a skewed normal to the eventual curvatures of a standard normal distribution. The curvatures around a standard normal appear as a donut or, a torus. Those curvatures are the same all the way around the normal in a 3-D view. That same donut around a skewed normal appears as a deformed donut, or a ring cyclied. In the skewed normal the curvatures differ from one side to the other. These curvatures differ all the way around the donut.

The curvature donut around the standard normal sits flatly on the x-axis and touches the inflection points of the normal curve. Dropping a line from the inflection points down to the x-axis provides us with a point where a line 45 degrees above the x-axis is where the origin of the circle of the particular curvature would be.

The curvature donut of a skewed normal would sit flatly on the x-axis, but might be tilted as the math behind a ring cyclied is symmetrical to another x-axis running through the centers of the curvatures. In January’s Kurtosis 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 smooth.

So I’m trying to overlay a 2-D view of a skewed distribution on a 3-D view of ring cyclied.

I’ve used a red line to represent the distribution. The orange areas are the two tails of the 2-D view. The curvatures show up as yellow circles. The inflection points on the distribution are labeled “IP.” The core is likewise labeled although the lines should match that of the tilted mean.

I think as I draw these figures, so in this one, have a gray area and a black vertical line on the ring cyclied that are meaningless. Further, I have not shown the orientation of the ring cyclied as sitting flat on the x-axis.

The ring cyclied occurs when skewness and kurtosis occur. A normal distribution exhibits skewness and kurtosis occur when the sample size, N, is less than 36. When N<36, we can use the Poisson to approximate or estimate the normal. Now, here is where my product management kicks in. We use Poisson games in Moore’s bowling ally to model Moore’s process as it moves from the early adopter to the chasm. The chasm being the gateway to the vertical market that the early adopter is a member of. We stage gated that vertical before we committed to creating the early adopter’s product visualization.  We get paid for creating this visualization. It is not our own. The carried component always belongs to the client. The carrier is our technology and ours alone.

So let’s look at this tending to the normal process.

I was tempted to talk about dN and dt, but statistics kids itself about differentials. Sample size (N) can substitute for time (t). The differentials are directional. But, in statistics, we take snapshots and work with one at a time, because we want to stick to actual data. Skew and kurtosis go to zero as we tend to the standard normal, aka as the sample size gets larger. Similarly, skew risk and kurtosis risk tend to zero as the sample size gets larger.

The longer conic represents the tending to normal process. The shorter conic tends to work in the inverse direction from the normal to the skewed normal. Here direction is towards the vertex. In a logical proof, direction would be towards the base.

The torus, the donut associated with the standard normal, like its normal is situated in Euclidean space. However; the ring cyclide is situated in hyperbolic space.

An interesting discussion on twitter came up earlier this week. The discussion was about some method. The interesting thing is what happens when you take a slice of the standard normal as a sample. The N of that slice might be too small, so skew and kurtosis return, as do their associated risks. This sample should remain inside the envelope of the standard normal; although it is dancing. I’m certain the footprints will. I’m uncertain about the cores in the vertical sense. Belief functions of fuzzy logic do stay inside the envelope of the base distribution.

Another product manager note: that slice of the standard normal happens all the time in the technology adoption lifecycle. Pragmatism orders the adoption process. Person 7 is not necessarily seen as an influencer of person 17. This happens when person 17 sees person 7 as someone that takes more risk than they or their organization does. They are in different pragmatism slices. Person 17 needs different business cases and stories reflecting their lower risk willingness. These pragmatism slices are a problem in determining who to listen to when defining a product’s future. We like to think that we code for customers, but really, we code for prospects. Retained customers do need to keep up with carrier changes, but the carried content, the use cases and conceptual models of carried content rarely the changes. The problem extends to content marketing, SEO, ancillary services provided by the company, and sales qualifications. Random sales processes will collide with the underlying pragmatism structure. But, hey, pragmatism, aka skew and kurtosis, is at the core of problems with Agile not converging.

In terms of the technology adoption lifecycle, the aggregated normal that it brings to mind is actually a collection of Poisson distributions and a series of normal distributions. The footprint, the population of the aggregated normal does not change over the life of the category. Provided you not one of those to leave your economy of scale with a pivot. Our place in the category is determined in terms of seats and dollars. When you’re beyond having sold 50% of you addressable population you are in the late market. The quarter where you left the early market and entered the late market is where you miss the quarter and where the investors are told various things to paper over our lack of awareness that lost quarter was predictable.

If you know anything about the ceiling problem, the sample distribution reaching beyond the parent normal let me know.

I’ve actually seen accounting visualizations showing how the Poissons tend to the normal.

Enjoy.