Lab Notebook: Why Sample Means Go Bell-Shaped
Sample means concentrate around the true mean with spread σ/√n because summing independent values convolves their distributions, an operation that erases sharp edges and corners until only a Gaussian profile remains.
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If the underlying data has no bell shape, why does the distribution of its sample mean look Gaussian?
Your model evaluates A/B tests on daily-active-user counts that look nothing like a bell curve — yet the averages you feed it behave as if they do.
A perfectly flat die has no peak. A wildly skewed revenue distribution has a sharp edge. So where does the smooth bell come from when you average them?
Run sample-mean simulations from a uniform source and a skewed exponential source at increasing n, then animate the convolution chain that smooths sharp edges into a Gaussian.
The bell curve is not inherited from the data — it is manufactured by averaging. Convolution smooths any source toward a Gaussian of width σ/√n, so the source shape stops mattering once n is large enough.
The shape of sample means must reflect the shape of the source data, so averages from a flat die should stay flat and averages from a skewed revenue stream should stay skewed.
- Formal proofs of the Central Limit Theorem
- Confidence-interval construction procedures
- Heavy-tailed distributions beyond a brief boundary mention
- Hypothesis testing or p-value interpretation
- 01Entry 01 — A Flat Die, a Bell Curve?slide
- 02Commit Before You Simulatequiz
- 03Notebook Cell — Sample Means from a Uniform Sourceinteractive
- 04Notebook Cell — Now With a Skewed Revenue Streaminteractive
- 05Why the Bell Appearsslide
- 06Notebook Cell — Watch Convolution Smooth the Edgesinteractive
- 07Margin Note — When the Bell Refuses to Formslide
- 08Notebook Cell — Bootstrap Your Own Datasetinteractive
- 09Entry 02 — The Bell Is Manufactured by Averagingslide
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