Run the CLT sandbox on a strongly skewed parent, and predict — before you read it off the screen — the standard error of the sample mean at one sample size, how that error changes when you quadruple n, and how the skewness of the sampling distribution shrinks as n grows.
The Central Limit Theorem says the sampling distribution of the mean tends to a normal, no matter how skewed the parent is — but that is a limit, and two numbers say how fast it bites. Its spread is the standard error SE = σ/√n, which falls like 1/√n: to halve it you must quadruple n, which is why precision is expensive. Its shape is what goes normal, and the honest measure of that is the skewness of the sampling distribution falling toward zero — for an i.i.d. mean it is γ₁/√n. Convergence is a number you can watch, not a vibe.
Open the sandbox. It opens on the Exponential parent with the Mean statistic — a strongly skewed parent, the case that embarrasses the “n ≥ 30” rule. Each step names the one knob to change; leave everything else at these values.
Parent -> Exponential (rate 1) Statistic -> Mean
n -> 4 Repetitions R -> 2500 Seed -> 20260821
Watch the spread (measured SD) readout with its σ/√n note, the n-knob derivation line that spells out the quadrupling, and the skewness readout in the stats panel. Every number you predict is printed there.
Set n = 4. The Exponential(rate 1) parent has σ = 1, so the standard error of the sample mean is SE = σ/√n = 1/√4. Predict it, then read the spread (measured SD) readout and its σ/√n note.
The σ/√n note reads 0.5. Even from a wildly skewed parent, averaging just four draws already pulls the spread of the mean down to half the parent's — and the measured SD of the sampling distribution sits right on that theory value.
Now drag n from 4 up to 16. Because SE = σ/√n, multiplying n by four divides the error by √4 = 2. Predict the ratio SE(16)/SE(4), then read the n-knob derivation line, which spells the factor out.
The factor is 0.5 — quadrupling n (4 → 16) exactly halves the standard error, taking it from 0.5 to 0.25. This is the 1/√n law, and it is why buying twice the precision costs four times the data.
Keep n = 4, repetitions R = 2500, seed 20260821 — the sandbox defaults. The parent's own skewness is 2; at n = 4 the sampling distribution still carries most of it. Read the skewness readout in the stats panel.
The skewness readout shows about 0.9168 — still markedly right-skewed, nowhere near a symmetric bell. Averaging four draws has cut the parent's skewness of 2 down by roughly √4, but four is far too few for this parent to look normal.
Now drag n up to 80, leaving R = 2500 and the seed alone. Skewness of the sample mean falls like γ₁/√n. Predict roughly where it lands, then read the skewness readout again.
The skewness readout drops to about 0.2521 — from 0.9168 at n = 4, the sampling distribution is now close to symmetric. That fall is the Central Limit Theorem made into a number: same skewed parent, but the mean of enough draws is nearly normal.
Everything above is waiting in the sandbox. Switch parents — Uniform, Bernoulli, Bimodal — and watch each one's mean go normal at its own pace; step the n ladder in panel E to find the first n that looks normal; then pick the Cauchy parent and the maximum statistic to see the two cases where the CLT does NOT apply, and the page withdraws its theory curve rather than lie.
- Read the standard error of the sample mean at n = 4: SE = σ/√n = 0.5
- Saw quadrupling n (4 → 16) halve the error, the 1/√n law
- Measured the sampling distribution's skewness at n = 4: about 0.9168 — still visibly skewed
- Watched it fall to about 0.2521 at n = 80 — the CLT as a number, not a vibe
- Every value you predicted is the demo's own sampling arithmetic, not a picture