Deeper

Lab · c12.s04.L1

The central limit theorem

Averaging independent draws replaces the source distribution with the same standard normal once the average is measured in units of sigma over the square root of n.

Question: Why does the same bell-shaped curve appear when you average many independent draws, whatever distribution the draws came from?

The central limit theoremOne chart per source, chosen from four: a lopsided coin, a fair die, waiting times and two clumps. A gold filled curve is the distribution of the average of n draws, from 4,000 simulations; a navy dashed standard normal stays fixed. At n = 1 the gold curve copies its source, with two spikes, six teeth, a long right tail or two humps. As n steps to 100, the gold curve settles onto the bell for every source.The central limit theoremn steps 1, 2, 3, 5, 10, 30, 100, one frame each; the axis is z = (average − μ) / (σ/√n)gold: distribution of the averagenavy dashed: standard normaldensity, same vertical scale in every frame-4.0-2.002.04.0n = 1 spread of the raw average = 0.458 (source units)source mean = 0.300 source spread = 0.458gold: standardised density of 4,000 simulated averages, bandwidth 0.15; navy dashed: standard normal
Re-simulated Re-simulated: 4,000 averages of n draws are simulated at n = 1, 2, 3, 5, 10, 30 and 100 for each of four sources by data/scripts/c12.s04.L1.py, and smoothed into densities with bandwidth 0.15 on a shared grid. The frames step n for one source, with the standard normal drawn fixed in navy; the gold curve moves onto it, so the same bell is reached from a coin, a die, waiting times and a two-clump source.
Lab controls
Coin, p = 0.3

Arrow keys move to the previous or next source; Home and End jump to the first and last

1100 ms per frame

Arrow keys change the interval; it does not change the frame

Frame 1 of 7: Coin, p = 0.3 — n = 1n = 1: the average of one draw is that draw, so the gold curve copies the coin. It has a tall spike for the common outcome and a short spike for the rare one, with empty space between them. Nothing here looks like the navy bell.

Keyboard: Space plays or pauses, Left and Right step, Home goes to the first frame, End goes to the last frame.

Text description of this lab

One chart per source, chosen from four: a lopsided coin, a fair die, waiting times and two clumps. A gold filled curve is the distribution of the average of n draws, from 4,000 simulations; a navy dashed standard normal stays fixed. At n = 1 the gold curve copies its source, with two spikes, six teeth, a long right tail or two humps. As n steps to 100, the gold curve settles onto the bell for every source.

Return to the section that hosts this lab

Caption status: re-simulated. Every lab declares one of published, re-simulated, schematic, synthetic or interpolated.