Take a population that is nothing like a bell — a hard floor at zero and a long tail, or nothing but zeros and ones — and average it. Drag the sample size and watch the lopsidedness drain out of the sample mean until what is left is a normal curve. Everything below is simulated and measured, not asserted: 4,000 studies are run at every setting.
Right-skewed populationn = 1 · drag the sample size to begin
Pick a population to average
Drag the sample size — this is the whole experiment
1
Population skew γ₁2.00
Predicted skew of x̄2.00γ₁/√n
Measured skew of x̄—from 4,000
Gap from normal D—noise floor .014
Does the 5% really mean 5%?
The picture is only half the story. If you use the normal model, you are betting that a stated probability is the real one. Each tile compares what the normal model promises with what actually happened in the 4,000 simulated studies.
Studies with z > +1.645—normal says 5.00%
Studies with z < −1.645—normal says 5.00%
Studies with |z| > 1.96—normal says 5.00%
Averaging is a shape-destroying operation. A single observation carries the population's whole personality — its floor, its long tail, its lumps. Add several together and the extremes have to conspire to survive: one huge value gets diluted by the ordinary ones beside it. What comes out the other side keeps only two facts about where it came from — the center μ and the spread σ/√n — and forgets the rest. The population's shape is not inherited by x̄. It is averaged away.
Read the theorem carefully — four things it does not say.(1) Your data do not become normal. The CLT is about the sampling distribution of x̄ — the pattern the mean would trace out over repeated studies. A sample of 5,000 from a skewed population still has a skewed histogram, and always will. Panel 1 above never moves.
(2) “n ≥ 30” is a rule of thumb, not a theorem. The theorem describes the limit as n → ∞ and promises nothing at any particular n. How fast the bell arrives depends on the population: n = 30 is generous for the flat one and badly insufficient for the extreme one.
(3) It needs conditions. Observations must be independent, and the population must have a finite variance. A population with tails heavy enough that σ does not exist — the Cauchy is the standard example — never converges to a bell at any n.
(4) We divided by the true σ here. In real work σ is unknown and you divide by the sample's own s instead. That adds a second source of randomness on top of everything shown above, and it is the entire reason the t‑distribution exists.