A real kick to the head.
One might wonder what the inherent value of a mathematical formula that can quantify horse kicks to the head might have in the real world.
1.
The mathematician Siméon Denis Poisson was renowned for his intellectual rigor, if a bit eccentric. He had a peculiar relationship with the number 7 and would schedule important events or start significant projects on dates related to this number. Notorious for correcting others, even in public settings, there's an anecdote where he once corrected a church sermon as it was being given.
However, he is most famous for the Poisson Distribution, a statistical function Poisson developed in 1837 while studying the number of errors in past court cases. Poisson analyzed data from French courts, specifically focusing on the frequency of errors or misjudgments in court cases over a certain period.
The Poisson distribution is like a crystal ball for events that happen randomly but with a known average rate. It doesn't give you the exact future but tells you how probable different outcomes are.
The math behind the formula is not too difficult, but I'll spare you everything but the most obscene generalities about the recipe. You'll need to preheat your oven to 450 and check your cupboard for these ingredients:
Lambda (λ) - your average rate.
The number of events you're curious about (like 3 notifications).
A constant (a type of special number in math)
A factorial (just multiplying numbers down to 1)
In the late 19th century, a Prussian military statistician named Ladislaus Bortkiewicz analyzed data from the Prussian army concerning the number of soldiers killed by horse kicks between 1875 and 1894. He looked at 14 corps over 20 years, with 280 observations. This observation became one of the earliest and most famous applications of the Poisson Distribution, also known as the 'Prussian Horse Kicks Data.'
Bortkiewicz was interested in how many soldiers in each corps died from horse kicks each year. Horse kicks leading to death were relatively rare, making it a perfect scenario for applying the Poisson distribution, which is excellent for modeling rare events. He found that, on average, there were 0.61 horse kick deaths per corps per year.
Using Poisson's formula there was a 54.3% chance of no one dying from horse kicks in any given corps in a year.
One might wonder what the inherent value of a mathematical formula that can quantify horse kicks to the head might have in the real world, but I can assure you Poisson reaches into your life constantly. Let's take a real-world example, your local MacDonald franchise:
If the average customer arrival at a drive-thru during lunch is 20 per 15 minutes (λ = 20), they can calculate the probability of 25, 30, or even 5 customers arriving in that time frame and adjust staff accordingly.
If on average, 150 burgers are sold per hour (λ = 150), they can use Poisson to estimate the probability of selling more or fewer than expected, helping with inventory control to avoid stockouts or overstocking.
In a well-managed McDonald's customer feedback or complaints are at an average rate with typical patterns or spikes. Variations likely indicate issues with service or product quality.
The Poisson distribution can model real-world phenomena where events occur independently of each other at a known average rate, even in something as unexpected as horse kicks leading to fatalities.
2.
What we call a bell-shaped curve is also known as a Gaussian Distribution, named after Carl Friedrich Gauss. When you plot this function, you get a curve that looks like a bell due to its smooth, symmetrical rise to a central peak and then a decline back to zero on both sides. The extremities of the curve are called the curves' tails (Indo-European *doklos, 'something long and thin'). When there is a higher probability of extreme values compared to what would be expected in a normal Gaussian distribution, these are called fat tails.
A perfectionist like Poisson, Gauss did not seem to have Poisson's need to correct everybody and was quite reticent, with a tendency to avoid disputes. Gauss was superstitious about the number 17 and reportedly discovered a method for constructing a 17-sided polygon with a ruler and compass, which was an achievement of great mathematical significance.
Many phenomena in nature and society can be approximated by a Gaussian distribution due to something called the Central Limit Theorem. If you take lots of samples and average them, no matter how weirdly the original numbers are spread out, the averages will look like they're coming from a bell-shaped curve.
Bell-shaped curves in general do a good job of modeling most natural occurrences. Their utility, use, and misuse, however, is legendary. As we have seen, things such as income distribution, city population sizes, or the frequency of word usage in languages tend to follow power-law distributions rather than Gaussian ones.
Gaussian models don't like outliers, which often is where the cool, interesting stuff is.
The concept of the black swan event, popularized by Nassim Nicholas Taleb, illustrates the impact of highly improbable, fat-tail events, with extreme consequences that aren't as improbable as they would appear when depicted inside a bell-shaped curve. The possibility of these black swan events should prompt crisis and emergency managers to always 'imagine the unimaginable.'
Putting too much faith in averages and bell-shaped curves can have negative effects on your health.
For example, a 5' 6" person who does not know how to swim should probably not try to ford a river with an average depth of four feet. Given a flat bottom, a conservative estimate would be that around 7% of the river depth would be over their head.