Draw any distribution — lumpy, skewed, two-humped, whatever. Then average a handful of random draws from it, over and over. Watch what the averages do.
1 / 4
Source distribution — paint it
drag across the box to draw
presets:
1
Distribution of the sample means
— theoretical bell overlaid once it appears
Trials run
0
Source mean μ
—
σ = —
Spread of means
—
measured σ̄
Predicted σ/√N
—
The aha: at N = 1 the blue histogram is just a copy of the shape you drew — averaging one number changes nothing.
Nudge N upward and no matter how jagged your source was, the averages pile into a smooth
bell curve, centered on the same mean and narrower by exactly 1/√N.
That universality — weird inputs, Gaussian output — is the Central Limit Theorem, and it's why the bell curve shows up everywhere.