Statistics
Central Limit Theorem
Draw a few numbers from any lopsided distribution and average them. Do it again and again: the averages pile into a bell, whatever the source looked like. The middle wins because there are many more ways to land there, and every four times the draws halves the spread.
source distributionaverages of n drawspredicted bell
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Spread of the mean σ/√n—
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Chapter
Source distribution
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averages collected—
σ of the source—
predicted spread—
measured spread—
What to observe
- With one draw the pile is just the source, lopsided. Averaging only two already pulls in the extremes: both draws would have to be extreme together.
- Two dice make 7 in six ways and 2 in one way. With three dice the counts already trace a bell: the middle can be reached in far more ways.
- Switch the source to Bimodal: two humps with nothing in the middle, yet the averages of 30 draws pile up exactly in that empty middle.
- Each four times the draws halves the spread: σ/√n. To be twice as precise you need four times the data.
- With ±1 steps and the Sum view, each trial is a random walk and the sum is where it ends. The paths fan out inside ±√k.
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