The Average, or the Core

Tonight I ended up reading some of the Wolfram MathWorld discussion of the Heaviside Step Function among other topics.  I only read some of it like most things on that site because I bump into the limits of my knowledge of mathematics. But, the Heaviside step function screamed loudly at me. Well, the figure did, this figure.

Cute

Actually, the graph on the left. The Heaviside step function can look like either depending on what one wants to see or show.

The graph on the left is interesting because it illustrates how the average of two numbers might exist while the reality at that value doesn’t. Yes, I know, not quite, but let’s just say the reality is the top and bottom line, and that H(x)=1/2 value is a calculated mirage. All too often the mean shows up where there is no data value at all. Here, the mean of 0 and 1 is (0+1)/2. When we take the situation to involve the standard normal, we know we are talking about a measurement of central tendency, or the core of the distribution. That central tendency or core in our tiny sample is a calculated mirage. “Our average customer …” is mythic, a calculated mirage of a customer in product management speak.

Cute w Nomal

Here I put a standard normal inside the Heaviside step function. Then, I show the mean at the x=1/2 of the Heaviside step function. The core is defined by the inflection points of the standard normal.

The distribution would show skew and kurtosis since n=2. A good estimate of the normal cannot be had with only two data points.

More accurately, the normal would look more like the normal shown in red below. The red normal is higher than the standard normal. The height of the standard normal shown in blue is around 4.0. The height of the green normal is about 2.0. The red normal is around 8.0. I’ve shown the curvature circles generated by the kurtosis of the red distribution. And, I’ve annotated the tails. The red distribution should appear more asymmetrical.

more acurately

Notice that the standard deviations of these three distributions drive the height of the distribution. The kurtosis clearly does not determine the height, the peakedness or flatness of the distribution, but too many definitions of kurtosis define it as peakedness, rather than the height of the separation between the core and the tails. The inflection points of the curve divide the core from the tail. In some discussions, kurtosis divides the tails from the shoulders, and the inflection points divide the core from the shoulders.

To validate a hypothesis, or bias ourselves to our first conclusion, we need tails. We need the donut. But, before we can get there, we need to estimate the normal when n<36 or we assert a normal when n≥36; otherwise, skew and kurtosis risks will jerk our chains. “Yeah, that code is so yesterday.”

And, remember that we assume our data is normal when we take an average. Check to see if it is normal before you come to any conclusions. Take a mean with a grain of salt.

Convolution

Another find was an animation illustrating convolution from Wolfram MathWorld “Convolution.” What caught my eye was how the smaller distribution (blue) travels through the larger distribution (red). That illustrates how a technology flows through the technology adoption lifecycle. Invention of a technology, these days, starts outside the market and only enters a market through the business side of innovation.

The larger distribution (red) could also be a pragmatism slice where the smaller distribution (blue) illustrates the fitness of a product to that pragmatism slice.

convgaus

The distributions are functions. The convolution of the two functions f*g is the green line. The blue area represents “the product g(tau)f(t-tau) as a function of t.” It was the blue area that caught my eye. The green line, the convolution, acts like a belief function from fuzzy logic. Such functions are subsets of the larger function and never exit that larger function. In the technology adoption lifecycle, we eat our way across the population of prospects for an initial sale. You only make that sale once. Only those sales constitute adoption. When we zoom into the pragmatism step, the vendor exits that step and enters the next step. Likewise when we zoom into the adoption phase.

Foster defined disruption as the interval when a new technology’s s-curve is steeper than the existing s-curve, we can think of a population of s-curves. The convolution would be the lessor s-curves, and the blue area represents the area of disruption.  Disruption can be overcome if you can get your s-curve to exceed that of the attacker. Sometimes you just have to realize what was used to attack you. It wasn’t the internet that disrupted the print industry, it was server logs. The internet never competed with the print industry. Fosters disruptions are accidental happenings when two categories collide. Christensen’s disruptions are something else.

Enjoy.

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