The Increase in the Decrease (Gaussian Distribution)
The probability of a deviation from the mean in a Gaussian distribution declines exponentially, with the odds decreasing dramatically with each additional unit of deviation. This accelerating decline allows statisticians to effectively ignore extreme outliers.
THE IDEA
The Increase in the Decrease
The main point of the Gaussian, as I've said, is that most observations hover around the mediocre, the average; the odds of a deviation decline faster and faster (exponentially) as you move away from the average. If you must have only one single piece of information, this is the one: the dramatic increase in the speed of decline in the odds as you move away from the center, or the average. Look at the list below for an illustration of this. I am taking an example of a Gaussian quantity, such as height, and simplifying it a bit to make it more illustrative. Assume that the average height (men and women) is 1.67 meters, or 5 feet 7 inches. Consider what I call a unit of deviation here as 10 centimeters. Let us look at increments above 1.67 meters and consider the odds of someone being that tall.*
10 centimeters taller than the average (i.e., taller than 1.77 m, or 5 feet 10): 1 in 6.3
20 centimeters taller than the average (i.e., taller than 1.87 m, or 6 feet 2): 1 in 44
30 centimeters taller than the average (i.e., taller than 1.97 m, or 6 feet 6): 1 in 740
40 centimeters taller than the average (i.e., taller than 2.07 m, or 6 feet 9): 1 in 32,000
50 centimeters taller than the average (i.e., taller than 2.17 m, or 7 feet 1): 1 in 3,500,000
60 centimeters taller than the average (i.e., taller than 2.27 m, or 7 feet 5): 1 in 1,000,000,000
70 centimeters taller than the average (i.e., taller than 2.37 m, or 7 feet 9): 1 in 780,000,000,000
80 centimeters taller than the average (i.e., taller than 2.47 m, or 8 feet 1): 1 in 1,600,000,000,000,000
90 centimeters taller than the average (i.e., taller than 2.57 m, or 8 feet 5): 1 in 8,900,000,000,000,000,000
100 centimeters taller than the average (i.e., taller than 2.67 m, or 8 feet 9): 1 in 130,000,000,000,000,000,000,000
… and,
110 centimeters taller than the average (i.e., taller than 2.77 m, or 9 feet 1): 1 in 36,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000.
Note that soon after, I believe, 22 deviations, or 220 centimeters taller than the average, the odds reach a googol, which is 1 with 100 zeroes behind it.
The point of this list is to illustrate the acceleration. Look at the difference in odds between 60 and 70 centimeters taller than average: for a mere increase of four inches, we go from one in 1 billion people to one in 780 billion! As for the jump between 70 and 80 centimeters: an additional 4 inches above the average, we go from one in 780 billion to one in 1.6 million billion* — Figure unavailable in the source files.
This precipitous decline in the odds of encountering something is what allows you to ignore outliers. Only one curve can deliver this decline, and it is the bell curve (and its nonscalable siblings).
When to use it
Understanding why extremely tall or short individuals are rare in a population.
In quality control, knowing that defects far from specification are extremely unlikely under a normal process.
In finance, using Gaussian models to price options assumes extreme moves are rare, which may be misleading in fat-tailed markets.
In scientific research, when analyzing measurement errors, the Gaussian assumption allows ignoring large outliers.
FROM THEORY TO EVERYDAY LIFE
In practice
Height Distribution Example
Using height as a Gaussian quantity: average 1.67 m. The odds of being 10 cm taller are about 1 in 6.3, but being 60 cm taller has odds of 1 in 1 billion, and 70 cm taller gives odds of 1 in 780 billion. Probability declines exponentially with each additional deviation.
A MOMENT TO REFLECT
Put it to work.
Think of a real-world quantity that follows a Gaussian distribution. How would the probability of extreme values change as you move further from the mean?
Why is it important to recognize that in many domains, extreme events are much rarer than expected under a Gaussian assumption?
Can you identify situations where people incorrectly assume a Gaussian distribution when the actual distribution has fatter tails?
CONTINUE THE CONVERSATION
Source texts
- The Black Swan The Impact of the Highly Improbable With a new section On Robustness and Fragility (Nassim Nicholas Taleb)