difficulty 3/5
Central limit theorem
Add up enough independent values from almost any distribution and the sum is approximately normal. Almost is doing real work — the theorem has conditions, and heavy-tailed data breaks them.
Intuition
The normal distribution turns up everywhere, and the usual explanation — that nature likes bell curves — explains nothing. The central limit theorem is the actual reason, and it is a statement about addition rather than about nature.
Take values from some distribution, any shape at all, and add them up. The sum does not look like the thing you started with. It looks like a bell, and the more terms you add the more exactly it does. Whatever asymmetry or lumpiness the original had gets averaged away, because for the sum to land far from centre, many independent values must all be extreme in the same direction, and that is rare.
That is why the theorem "explains the prevalence of normal distributions in the natural world": many measured characteristics "are the sum of a large number of small effects, so their distribution tends to be normal" [1, §14.4].
It is also why the word almost in "almost any distribution" deserves attention. Some distributions do not comply, and the ones that do not are exactly the ones that turn up in revenue, file sizes and network traffic.
Formal definition
How it works
Where it's used
Why & when
Worked example
Pitfalls
Check yourself
3 questions
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On the cheat sheet: Statistical testing →
Sources
- Think Stats, 2nd ed.book · retrieved 2026-09-06https://greenteapress.com/thinkstats2/html/index.html
- NIST/SEMATECH e-Handbook of Statistical Methodsstandard · retrieved 2026-09-06https://www.itl.nist.gov/div898/handbook/eda/section3/eda3661.htm