Central Limit Theorem

Equivalent forms

\bar{X}_n\sim\mathcal{N}\!\left(\mu,\tfrac{\sigma^2}{n}\right)

The reason the bell curve rules nature: sums of independent randomness forget their origins and converge to one universal shape.

Symbol	Name	SI unit	Dimension	Range
$n$	Sample size Number of independent variables summed	1	1	1 – 50
$\mu$	Mean Expected value of each variable	1	1	—
$\sigma$	Std deviation Spread of each variable	1	1	0.1 – 3
$Z$	Standardized sumoutput Normalized sum converging to N(0,1)	1	1	—

Where it holds

Requires independent variables with finite variance; the classic form also assumes identical distributions. Fails for heavy-tailed variables (infinite variance → stable/Lévy limits instead).

Dimensional analysis

Z = (S_n - n\mu )/(\sigma \sqrt{n})

→ dimensionless

N(0,

1) \to

dimensionless

Subtracting the mean and dividing by the standard deviation cancels units, giving a pure number.

\checkmark

Discovery

de Moivre, Laplace, Lyapunov · 1810

de Moivre found the normal approximation to the binomial in 1733; Laplace generalized it around 1810. Lyapunov (1901) and later Lindeberg gave rigorous conditions. Pólya coined 'central limit theorem' in 1920 — 'central' meaning central to probability theory.

Research status: stable

Real-world applications

Justifying Gaussian error bars on averaged measurements
Statistical mechanics: macroscopic fluctuations are Gaussian
Signal averaging and noise reduction $(SNR \propto \sqrt{n})$

Common misconceptions

The CLT is about the distribution of the SUM/average, not the raw data
It needs finite variance — Cauchy-distributed data never converge to normal
Convergence speed depends on skewness; small n can be far from Gaussian

Experimental verification

Galton's bean machine (quincunx) physically builds a bell curve from many binary deflections; histograms of averaged noisy measurements confirm the

1/\sqrt{n}

narrowing daily.

Derivation

Standardize the

sum S_n = \Sigma X_i

and study its characteristic function

\varphi (t)=E[e^{itZ}]

.

For

Z=(S_n-n\mu )/(\sigma \sqrt{n})

, independence factorizes

\varphi

into a product; expanding each factor's log to second order (finite variance) gives

\varphi (t) \to e^{-t^{2}/2}

as

n\to \infty

.

Since

e^{-t^{2}/2}

is the characteristic function of N(0,1), by Lévy's continuity theorem Z converges in distribution to a standard normal.

What if…

What if the underlying distribution is very skewed?

The average still converges to normal, but you need larger n; the Berry–Esseen bound quantifies how slowly.

What if variance is infinite (heavy tails)?

The CLT fails; sums converge instead to a stable (Lévy) distribution — relevant to turbulence and finance.

1

Averaging noisy measurements

Given ·

σ:: single-shot noise
n:: 100 repeats

Find · uncertainty of the mean

Steps

Variance of the mean $= \sigma ^{2}/n$
$\sigma$ _mean $= \sigma /\sqrt{100} = \sigma /10$
$10\times$ more precision for $100\times the$ data

Result ·

\sigma

_mean

= \sigma /\sqrt{n} = \sigma /10

Binomial Distribution→law of large numbersgaussian integral