Plato's Curve (Gaussian Bell Curve)

FIGURE 10: A MORE ABSTRACT VERSION: PLATO’S CURVE

An infinite number of tosses.

Let’s keep going. We can flip 4,000 times staking a tenth of a penny. How about 400,000 times at 1/1000 of a penny? As a Platonic form, the pure Gaussian curve is principally what happens when we have an infinity of tosses per round, with each bet infinitesimally small. Do not bother trying to visualize the results, or even make sense out of them. We can no longer talk about an “infinitesimal” bet size (since we have an infinity of these, and we are in what mathematicians call a continuous framework). The good news is that there is a substitute.

We have moved from a simple bet to something completely abstract. We have moved from observations into the realm of mathematics. In mathematics things have a purity to them.

Now, something completely abstract is not supposed to exist, so please do not even make an attempt to understand Figure 10. Just be aware of its use. Think of it as a thermometer: you are not supposed to understand what the temperature means in order to talk about it. You just need to know the correspondence between temperature and comfort (or some other empirical consideration). Sixty degrees corresponds to pleasant weather; ten below is not something to look forward to. You don’t necessarily care about the actual speed of the collisions among particles that more technically explains temperature. Degrees are, in a way, a means for your mind to translate some external phenomena into a number. Likewise, the Gaussian bell curve is set so that 68.2 percent of the observations fall between minus one and plus one standard deviations away from the average. I repeat: do not even try to understand whether standard deviation is average deviation—it is not, and a large (too large) number of people using the word standard deviation do not understand this point. Standard deviation is just a number that you scale things to, a matter of mere correspondence if phenomena were Gaussian.

These standard deviations are often nicknamed “sigma.” People also talk about “variance” (related, but distinct: variance is the square of the sigma, i.e., of the standard deviation).

Note the symmetry in the curve. Equal deviations above and below the mean have equal probabilities. The odds of falling below −4 sigmas are the same as those of exceeding 4 sigmas, here 1 in 32,000 times.

As the reader can see, the main point of the Gaussian bell curve is, as I have been saying, that most observations hover around the mediocre, the mean, while the odds of a deviation decline faster and faster (exponentially) as you move away from the mean. If you need to retain one single piece of information, just remember this dramatic speed of decrease in the odds as you move away from the average. Outliers are increasingly unlikely. Whether they can be ignored depends on the consequences of an outlier and on whether the Gaussian model fits.

This property also generates the supreme law of Mediocristan: given the paucity of large deviations, their contribution to the total will be vanishingly small.

In the height example earlier in this chapter, I used units of deviations of ten centimeters, showing how the incidence declined as the height increased. These were one sigma deviations; the height table also provides an example of the operation of “scaling to a sigma” by using the sigma as a unit of measurement.