Bell Curve (Gaussian Distribution)
The bell curve, or Gaussian distribution, describes how many natural phenomena cluster around an average, with fewer instances at extremes. Sir Francis Galton's enthusiasm for this distribution, driven by its Platonic purity, helped spread its use into statistics and social sciences, though it is often misapplied.
THE IDEA
“The Greeks Would Have Deified It”
The list of people walking around with the bell curve stuck in their heads, thanks to its Platonic purity, is incredibly long.
Sir Francis Galton, Charles Darwin’s first cousin and Erasmus Darwin’s grandson, was perhaps, along with his cousin, one of the last independent gentlemen scientists—a category that also included Lord Cavendish, Lord Kelvin, Ludwig Wittgenstein (in his own way), and to some extent, our überphilosopher Bertrand Russell. Although John Maynard Keynes was not quite in that category, his thinking epitomizes it. Galton lived in the Victorian era when heirs and persons of leisure could, among other choices, such as horseback riding or hunting, become thinkers, scientists, or (for those less gifted) politicians. There is much to be wistful about in that era: the authenticity of someone doing science for science’s sake, without direct career motivations.
Unfortunately, doing science for the love of knowledge does not necessarily mean you will head in the right direction. Upon encountering and absorbing the “normal” distribution, Galton fell in love with it. He was said to have exclaimed that if the Greeks had known about it, they would have deified it. His enthusiasm may have contributed to the prevalence of the use of the Gaussian.
Galton was blessed with no mathematical baggage, but he had a rare obsession with measurement. He did not know about the law of large numbers, but rediscovered it from the data itself. He built the quincunx, a pinball machine that shows the development of the bell curve. True, Galton applied the bell curve to areas like genetics and heredity, in which its use was justified. But his enthusiasm helped thrust nascent statistical methods into social issues.
When to use it
Analyzing standardized test scores to identify how many students perform above or below a certain threshold.
Quality control in manufacturing: monitoring product dimensions to see if they fall within acceptable limits around a target value.
Understanding natural variability in biological measurements like blood pressure, weight, or leaf sizes.
Modeling measurement errors in experiments, where repeated measurements often cluster around the true value.
FROM THEORY TO EVERYDAY LIFE
In practice
Galton's Quincunx
Galton designed the quincunx, a pinball-like machine with rows of pins, to visually demonstrate how random drops of balls form a bell curve pattern. This illustrated the Central Limit Theorem and the emergence of the normal distribution from many independent random events.
Human Height Distribution
Within a comparable group of adults, heights may follow an approximately normal distribution: observations cluster around a mean, with fewer at either extreme. Combining groups with different means can produce a different shape.
A MOMENT TO REFLECT
Put it to work.
Why does the bell curve's 'Platonic purity' sometimes lead people to overapply it to situations where data follows a different distribution (e.g., power laws)?
How might the historical enthusiasm for the bell curve, as described by Galton, have influenced its use in social sciences and policy (e.g., intelligence testing)?
What are the key assumptions required for a phenomenon to follow a normal distribution, and how often do those assumptions hold in real-world data?
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)