The Central Limit Theorem

Sampling distributions

The Central Limit Theorem

Average enough draws from almost any population and the averages pile up in a bell curve — centred on the same place, narrower by a factor of √n, and normal no matter what the population looked like.

Three live simulations · normal, uniform, right-skewed and bimodal parent populations

The normal population

Thirty observations are drawn, collapsed to their mean, and dropped into the histogram on the right. Start here, where the population is already a bell curve and nothing needs proving.

Live simulation · 1,600 sample means from N(0.500, 0.120²), then it clears and runs again.

Populations that are not normal

Now take the symmetry away. These three join the normal population in everything that follows — flat, heavily right-skewed, two-humped, and not one of them bell-shaped. The theorem does not ask them to be.

Uniform density: perfectly flat from 0 to 1

Uniform

μ = 0.500   σ = 0.289

Right-skewed density: a tall spike near zero with a long tail to the right

Right-skewed

μ = 0.211   σ = 0.195

Bimodal density: two separate humps near 0.22 and 0.78

Bimodal

μ = 0.500   σ = 0.289

Densities estimated from 600,000 simulated draws of each population; all three live on the interval 0 to 1.

Averaging, one sample at a time

The same machine, now fed populations that are nothing like a bell curve. Both panels share one axis, so the narrowing is real, not a change of scale.

Live simulation · 1,600 sample means per population, cycling through the uniform, right-skewed and bimodal parents.

The role of sample size

The same right-skewed population, three sample sizes, one shared axis. Two things change at once: the skew straightens out, and the spread shrinks by a factor of √n.

Right-skewed population, μ = 0.211 and σ = 0.195 · 2,500 sample means per panel, then the simulation restarts.

The statement

X

≈ N(μ, σ²∕n)  as n → ∞

for X₁, X₂, …, Xn drawn independently from one population with finite mean μ and finite variance σ². The approximation is about the distribution of X̄ across repeated samples — not about any single sample.

Centre

E(X̄) = μ, whatever n is. Averaging never shifts the target.

Spread

SD(X̄) = σ∕√n, the standard error. Four times the data for half the error.

Shape

Approaches the normal curve regardless of the parent population's shape.