ZENTRALER GRENZWERTSATZ
The shape of chance.
Add independent random numbers. Repeat. Watch a bell curve emerge from surprisingly different beginnings.
02 / ADD, REPEAT, COMPARE
A sum becomes a shape
03 / INSIDE ONE TRIAL
One sum. Then do it M times.
Every histogram entry comes from a fresh set of N independent draws.
04 / THE BIG PICTURE
Different N. The same destination.
Standardized sums · 4,000 trials per panel
These independent simulations use your selected distribution and parameter. For discrete inputs, the bars stay discrete even as their envelope approaches a bell curve.
WHY IT WORKS
The center adds.
The spread grows more slowly.
For independent, identically distributed Xᵢ with mean μ and finite, positive variance σ², adding N terms gives these exact moments:
E[Y] = Nμ · Var(Y) = Nσ²
SD(Y) = √N σ
Centering and scaling gives Z = (Y − Nμ)/(σ√N). As N grows, its distribution converges to the standard normal distribution. The Gaussian is an approximation at finite N.
READING THE EXERCISE
Three starting points
| Input Xᵢ | Mean μ | SD σ |
|---|---|---|
| ±a, equally likely | 0 | a |
| Uniform [−b, b] | 0 | b/√3 |
| Poisson(λ) | λ | √λ |
The exercise’s ΔX and ΔY denote standard deviations, not variances. Pairwise zero correlation alone is not enough for the usual central limit theorem; this toy uses independent draws.
Try Poisson with λ = 0.2: small N gives a strongly skewed shape. Its sum is exactly Poisson(Nλ), so increasing Nλ makes the Gaussian approximation better.
What does the vertical axis mean?
Bars show probability density: count ÷ (M × bin width). Their total area is 1, including for discrete distributions, where lattice-aligned bins turn probability masses into comparable bar heights. Density can exceed 1. The curve is exp(−½((x−μY)/σY)²)/(σY√(2π)); in standardized view μY = 0 and σY = 1.