A tweet sent me to “Mean, Median, and Skew: Correcting a Textbook Rule.” The textbook rules are about the mean being in the long tail and the mode being in the short tail. The author discussed exceptions to this rule. Figure three presented me with a distribution that the author claims to be a distribution that was an exception to the textbook rules. The author claims the distribution is a binomial. I annotated the figure. It’s definitely some kind of a nomial, but looking closer, it is not a binomial.

The nominal on the right side of the distribution shows us what we see if we look at the side of any normal. An aggregate curve comprised of a concave downward curve and a concave upward curve with an inflection point between them, a single inflection point between them.

The distribution on the left side is not the result of a single nominal. There are many inflection points. The left side of the distribution is concave down, concave up, concave down, and concave up. We can say the left tail is single tail comprised of two presented lines, or we can say they are the overlap of two different distributions. That second concave down hides a distribution inside the base distribution.

The distribution gets called a binomial because it has two prominent peaks. But the left peak is an aggregate of at least one more nomial. Otherwise, we would add another set of inflection points. When making an argument about where the mean, median, and mode are we have to consider each nomial to have its own triple. So there should be at least two triples, rather than one, as shown in the figure. I called the triple we were presented with an error, but it does present us with one of the exceptions the author wants to talk about. From this, we can take away the idea that these aggregate statistics hide more than they inform. I found myself in a Quora discussion on separating the underlying distributions of a binomial. There is math for that, math I do not know yet.

I am working on the assumption that all the underlying distributions are normal, a base assumption that is routinely made in statistics.

The graph hides much as well so I drew what I expected the distributions under the given “binomial” would be. I just eyeballed it.

I used arrows that match the color of the curve to show the concavity. Extra probability mass shows up at the intersections where distributions meet. I’ve labeled the probability mass at the intersections as gaps. Given the underlying distributions are only approximations, I didn’t make the green distribution, distribution 1, fit perfectly, so the thin layer of the second gap from the beginning lays on top of the distribution without involving a distribution. I used three different distributions to account for the tail convergence on the right. This gave rise to a gap. I didn’t catch this when I drew the figure. As I write this, there is no gap there. The red distribution accounts for that probability mass.

I went with a skewed distribution, distribution 1, to account for the second concave down section of the curve on the left side of the “second” nominal. A normal wouldn’t bulge outward under the exterior nominal, the black normal. A skewed normal has a long tail and a short tail. The intrinsic curvature of any long tail is low, so it has a large radius. The intrinsic curvature of any short tail is high giving us a small radius. The mean of this distribution is to the left. The median pushes the mean and mode apart symmetrically about the median. The median for distribution 1 leans to the right.

I went with three peaks on the left side of the “binomial.” I did this because distributions 2 and 4 have different heights. I know of no rules that would drive this decision. They could easily be one distribution.

The rest of our “binomial,” actually as demonstrated, it is a multinomial instead. We’ve ended up with five distributions so we would have five different triples of mean, median, and mode. These triples were aggregated in the author’s numeric results. We can take it that when the mean, median, and mode are the same, we have a standard normal. The textbook rules about the tails and their relationships to the mean and mode still stand. Otherwise, we have numbers generated from an aggregate normal.

Don’t just accept the “binomial” allegation. If the numbers don’t make sense, they don’t make sense. When numbers don’t make sense, you’ve got more sense to make.

As a product manager, I don’t want to aggregate and drive that into a product that fits no one.

I went on to play with the “binomial” distribution some more.

I started with vertical slices for the Riemann integral. I also did this to give me a hint towards the factors involved in each slice. Due to my use of raster graphics, some slice lines are thick, because the intersections of the distributions are not points. Some intersections are lines. The point intersections give rise to vertical lines. The line intersections give rise to rectangles. Each vertical slice in those rectangles can differ. They are not uniform. Individual slices would still look like a solid rectangle.

The vertical lines tell us that at that moment in time, our organization if we worked at the underlying granularity, would represent some management adjustment to serve the underlying populations appropriately. This both the gray and light blue lines or rectangles.

The blue lines show us where the associated distribution converges with the horizontal axis. That horizontal axis would move relative to any upmarket or downmarket moves the organization was undertaking over a period of time. I labeled these as ordering changes. But, the gray lines are ordering changes as well. Orderings come up when computing binomial probabilities and in game theory.

The pink area shows the expanse of a single factor mixture. Part of that area shows the factor associated with the black distribution quickly slowing down. I labeled that part of the black curve “Fast.” And, it shows the factor’s deceleration showing. That labeled “Slow.” Otherwise, this slice is relatively stable. Note growth is not a positive notion here. In fact, the late phases of the technology adoption lifecycle, the orthodox management phase is post growth and in decline–constant decline. The only options are to focus, an upmarket move, or to drop the price and move downmarket. Neither guarantee growth in themselves.

From the mean of distribution 5, the purple distribution, All factors are in decline. But in the pink area, the factors are organized by a single constant factor curve.

In Upton’s *“Aesthetics of Play”*, the pink zone is a single play space. In his book, rules generate spaces and those spaces dictate process and policy. The technology adoption lifecycle(TALC) is based on this idea, but it is based on populations organized by that population’s pragmatism. The business facing that play space or population must eliminate its process and policy impedances to succeed. Addressed impedances constitute your organization’s design.

These spaces make those nascent moments when we don’t have a normal part of the difficulty with bringing another discontinuous innovation to market while sitting in the space where the category the company is in is dying. The pink space is that end-of-life space. Notice how different the pink space is to any slice on the left side of the aggregate distribution.

Upmarket and downmarket moves move the feet of the distributions, the points of convergence with the horizontal. The new space might have additional intersections of the nominal distributions. Where this is the case, the factors for the new slices would change. This would repartition the existing populations as well. Where the nominals are normal, the additional populations gained by the move would not change the nominals other than at the feet. In upmarket moves, keep them large enough to maintain normality, or expect exposure to kurtosis risk.

In our diagrams, the red distribution seems high, which implies that it needs more density. The number of data points needs to be increased. This also implies that there should be some skew, but it is not apparent. As a distribution gains probability mass, it becomes lower and wider.

When looking for inflection points, those points can be lines. The nominal on the right exhibit that behavior. I went looking for what that means mathematically. The inflection point is ambiguous. I crossed paths with symplectic geometry. They deal with the same problem. The nice thing businesswise about this ambiguity is that it grants you some time to switch from growth to decline or from fast to slow. The underlying processes of the business need to change at all inflection points. The deal here between a point and a line is that a point is a sudden change requiring proactivity, and a line requires less proactivity.

Then, I wanted to see the toruses involved. So I started with the normal distribution on the right side of the “binomial.” I used the original distribution, not the teased out distribution, so the distribution on the left only exposed its left side. fitting a circle to the curve on the left was less clear.

Imagine if a tori pair was shown for each of the five distributions. Where a tori pair does not have the same radius in each constituent circle, there would be kurtosis, a pair of tails, and a median lean. The radii of the circles in that pair would change as the 2D slicings were rotated around the underlying distribution. The median lean results from the particular dimensions of the 2D slice. This generates some ambiguity in the peak, as the median for each slice would differ. By slicings, I mean taking slices around the circle giving us a collection of different slices. I do not mean rotating the same slice.

Where a tori pair had the same radius, the distribution has achieved normality. The kurtosis would be near zero, the median would no longer lean, and the mean, median, and mode would converge to the same value. The radii of the circles would not change as the 2D slicings were rotated.

Next, I took horizontal slices as in Lebesgue integrals.

As discussed in regards to the vertical slicing, the gray lines indicate point intersections. The thicker gray lines indicate line intersections.

Where the vertical slice figure showed gaps, those gaps are comprised of a collection of Poisson distributions and a single collective normal. Poisson distributions come to approximate the normal when it has 20 or more data points. The normal is achieved without approximation when 36 data points have been collected. Breaking a normal into subsets can give rise to Poisson distributions. So there is risk involved with these considerations. I highlighted these with yellow rectangles around the labels.

The skewed distribution, the green distribution, has been highlighted with the same yellow as the Poisson distributions because having not yet achieved normality, much will change and those changes will be rapid as normality is achieved.

The red arrows show the direction in which I expect the distribution to change. The left arrow associated with the skewed distribution is only considering the movement of the foot, everything will change with the skewed distribution. The base “binomial” will most likely change and give rise to an apparent 3rd nominal on the exterior of the aggregate distribution. The down arrows associated with the peaks can be expected to lose height or amplitude as more data is collected.

The median of the skew would become orthogonal. The change in its theta is not indicated on the diagram.

The intersections of the distributions will change, so they are highlighted in yellow as well.

The factor analyses also change when looked at from a horizontal slice point of view. You can consider the factors across a horizontal slicing to differ from the factors across a vertical slicing. There would be a collection of cubes if both slices where made. Those cubes would be N-dimensional, but given our slicings would be 2D, it would get messy. cubing based on a factor analysis would be easier to operationalize in the sense of organizational design.

I labeled the slices. I had intended to provide a factor analysis for each slice. If I had the underlying data that would have been possible, but a graphical approach proved frustrating.

Next, I generated the probability of a portion of the AI slice under distribution 5, the purple distribution. A Lebesgue integral would achieve the same result.

The blue rectangle represents the probability mass under the purple distribution between the vertical constraints of the gray lines delineating that dimension of the slice AI.

The author went on to give several examples of other aggregate distributions. He used these distributions to explore how the mean, median, and mode violate our expectations. So the textbook rules are violated by aggregates of underlying distributions, multiple distributions. This is true of the “binomial” example. As a rule, only consider those statistics to be valid at the level of the constituent nomials, rather than the aggregate nominal. Aggregate nominals frustrate the expected orderings of the statistical tuples.

I take it that the thick black line is the mode. On the left, we get the textbook ordering. Then, in the yellow rectangle to the right of 0.5, it changes to an exceptional ordering. At some point, it changes back to textbook ordering. And to the right of 0.75, the mean changes its tail association to being associated with the short tail. In the textbook ordering the mean is in the long tail. This is where using a single number for kurtosis does not make sense. It only made sense in the standard normal sense where the tails have identical values on both sides on the 2D slice involved.

The author went on to construct a distribution associated with the graph showing the tuple ordering exceptions. In a skewed normal, the median leans over to sit on top of the mode. This is the case in the aggregate distribution used here. The ordering is not exceptional, but the lean is not at the value of mode but along it. Where I annotated this as exceptional, the exception is the distance from the median to the mode. The ordering is not exceptional. It does, however, change the width of the separation between the median and the mode. The ordering is not symmetric around the median. The red lines are intended to show the median leaning on the mean so that the asymmetry relative to the mean, median, and mode is clear.

Then, I went on to explore the logic of the 2D slice. Here we are talking about the logic of the carried data, not the logic of the statistical carrier. The logic of the statistical carrier would be that of a normal distribution. With all the mathematical approximation formulas allowing us to convert from one distribution to another, we might ignore the logical constraints. I’m calling these distribution-to-distribution logical constraints the logic of the statistical carrier. The aggregation rules for a normal is an example of such carrier constraints. The carried logic is that of the collected data, rather than the collection and analysis of such data.

Logical consistency is tricky. Decades ago consistency was a true or false question. Was it consistent from the top to the bottom across every branch of the argument? These days that’s called absolute consistency. But now, we have relative consistency. It works from some absolute consistency to a branch of the argument that is consistent with itself and that base absolute consistency. Other branches would arise. Those branches would not demonstrate absolute consistency with other branches. This kind of consistency is relative consistency.

Statistically, the relative consistency would be a characteristic of each tail. Absolute consistency would be a characteristic of the core.

Relative consistency leaves us in a non-Euclidean space. That space typically would be hyperbolic involving manifolds, rather than functions. This calls into question the management practice of alignment and organizational structure.

In this figure, the logic of the tails is highlighted in pink. The question marks indicate where one would define shoulders, outliers, and distant outliers. What are your definitions of those boundaries? This is a 2D slice. Another 2D slice through the mean might require different decisions. Another slice would have a different set of curves. One of the slices would appear to be a standard normal with equal tails on both sides of its mean.

Relative consistency would start at the shoulder of a particular tail. Where you don’t differentiate the shoulders from the tails, a relative consistency starts with a particular tail. Each tail would have its own logic.

The last figure demonstrates the slices concept. The red line is closer to a standard distribution and its tails. The blue slice is definitely skewed. The thin blue line in the core is there to hint at the lean involved in that 2D slice. The red slice does not exhibit any lean. As more data of the dimension underlying the blue baseline is collected, the lean will disappear as will the asymmetry of the tails.

As a manager, big data is great if you have large existing populations and large existing collections of relevant data. Continuous innovation thrives in this situation. But, do be cautious of Poisson scale subsets. And, be cautious of any distribution summed to the existing normals. That data might be Poisson. And, that distribution would be skewed and kurtotic bringing you their relevant risks. Discontinuous innovation is blank space inventions tied to an absence of any relevant populations. These innovations have tiny networks. Data collected from those networks will be small data, Poisson, pre-normal, and will move across the terrain. It will be a long time before it settles down, but at the same time, it is a long way from being a commodity, or something that orthodox management practice can handle. It is a long way from the spherical geometry of that orthodoxy. It is a long way from the Euclidean of LP2. It is hyperbolic. All that distance implies there is real economic wealth to be created, and there is plenty of time to capture it.

The data collection and relevant distributions will mature.

Snapshot statistics is not all that informative. What your distributions dynamically.

Enjoy.