The Gaussian Fallacy

The most used model in management theory is the model of the bell curve, or a Gaussian distribution.  If one were to study for an MBA anywhere in the world, one of the first things taught would be statistical control based upon a bell curve.  Managers learn that the world consists of Gaussian distributions, and that one can safely ignore the occurrence of things or events past the second or third standard deviation.  In theory, if one understands the bell curve, they understand how the world works.  It’s an interesting reduction.  It’s clearly a way of reducing the world in order to achieve an illusion of the ability to cope, and …. it’s dead wrong.   

 

Assumptions — Bad Assumptions

The Gaussian distribution is based upon a distinct set of assumptions.  The first is that ascribed labels matter. If a label is assigned to something, the assignment will be accurate within two or three standard deviations, and that’s good enough.   The second assumption is that whatever we’re encountering in the world obeys the law of large numbers. This states that large numbers drift toward obeying the Gaussian distribution.  But, it is the third assumption that is the most critical. The real underlying assumption of a Gaussian distribution is that events are independent and truly random (in a mathematical sense).  We don’t live in that kind of world.   Rather, events are at least partially correlated, and usually not fully independent.  What we believe to be random is some illusion of randomness — the patterns do not match the mathematical definitions of true randomness.  As a result, random as the layman sees it is not really the random of mathematics on which the distribution depends, instead there are elements of connectedness and mathematical deviations from the true random. We misunderstand the law of large numbers to suggest that when we see noise, it’s noise.  Complexity theory suggests that “noise” might be a weak signal of something else.   The law of large numbers makes no room for weak signals.   The problem with ascribing a label, and using it as your method of explanation, is that once one has ascribed it, once one has said this belongs to Label X, then the explanation is done.  There’s no room in that ascription for emergence.  Yet, emergence happens. If you shift your scale, if you shift your context, if you encounter something new, one might spend a lot of time trying to make the old ascribed label fit.  If you are the “explanation is the assignment of a label” kind of person, you don’t spend time trying to understand the emergence that has just occurred.   

 

What is Wrong With Gauss?

 The graph above illustrates the problem.  Much of the time the middle section of the graph, the part all the way over on the right, should be smaller.   Again, much of the time, the tails of the graph actually do not fall off anywhere near as sharply as Gaussian distribution suggests, and should be bigger (the supposedly rare events are less rare than the distribution suggests). Outside events, which the Gaussian distribution suggests should be disregarded as noise, may have some validity and should be investigated.   The “average expected” middle, which the Gaussian distribution suggests will have that nice distribution around it, is also not quite as big as we are led to believe. 

They don’t teach this in an MBA program.  They should.

 

When the Gaussian Distribution Does Not Apply

When we observe events where the Gaussian distribution does not apply, what does that signify?  Is it noise?  Is it emergence?  Has one shifted scales?  Has one shifted context?  Is it a weak signal of something?  Is there partial dependence or partial correlation?  When the Gaussian distribution is inappropriate as a label, it’s a strong cue. If one learned statistical control, ala Edwards Deming, when Gauss is inappropriate, it’s a signal that there’s noise, and the system is broken.  But this may not be at all what the signal is.  Instead, when the Gaussian distribution fails to hold, it may signal that emergence is occurring, it may signal that the wrong questions are being asked, or it may signal that prior categories are breaking down.  The concept of “model” which we teach managers often fails to convey these lessons.