Central Limit Theorem

Take any distribution with a finite mean and variance. Draw $n$ independent samples from it and average them. Now repeat — thousands of times — and look at the distribution of those averages. No matter what you started with, that distribution gets arbitrarily close to a bell curve as $n$ grows.

\overset{ˉ}{X}_{n} = \frac{1}{n} i = 1 \sum n X_{i}

\overset{ˉ}{X}_{n} d N (μ, \frac{σ ^{2}}{n})

Source distribution:

Source

μ = 0.00, σ = 2.06

2000 sample means, each of n = 1

Curve: theoretical limit N(μ, σ²/n), with σ/√n = 2.062

n (sample size) 1 k (number of means) 2000

What to try

Start with n = 1. The histogram looks like the source — uniform is flat, exponential is skewed, bimodal has two humps.
Now drag n to 5. Already bell-ish, even for bimodal.
Push n to 30. Essentially Gaussian for all three.
Flip between sources while keeping n high — the histogram shape barely changes, only its width.

What to notice

The centre of the bell is always the source mean. Averaging doesn’t move it.
The spread is $σ / n$ — the source’s standard deviation divided by $n$ . Double n, narrow by $n$ .
You need a surprisingly small n. Textbooks quote 30 as a rule of thumb; for a well-behaved source, 10 is often enough.
For pathological sources — say, a Cauchy distribution with no defined variance — the theorem doesn’t apply and averaging doesn’t help.

Why it matters

This is why so much of classical statistics treats errors as normal: averages (sums) of many small effects are almost always approximately Gaussian, even when no individual effect is. The CLT is the bridge between messy real-world sampling and the clean theory of the normal distribution.