Interactive guide
Six wildly different starts. One shared ending.
Six independent running averages of the same population, on a log scale so early chaos and late calm both fit.
running average of one sequence (×6) true population mean
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- True mean
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- Avg |error| now
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- σ/√N (theory)
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Drag the slider back to N = 1 and every one of the six lines starts somewhere different, some near the true mean by luck, some far from it. Push it up toward 1000 and watch that spread of starting points stop mattering: however far off a sequence started, it ends up exactly as close to the true mean as all the others.
Not the same claim as the Central Limit Theorem
It's easy to blur these two together since both involve averaging many draws, but they answer different questions. The Central Limit Theorem says: the sampling distribution's shape becomes bell curve-shaped as n grows. The Law of Large Numbers says something plainer: the sampling distribution's center sits exactly on the true mean, and its spread keeps shrinking, so a single running average gets closer and closer to that one number, almost surely.
You could have a sampling distribution that stays lopsided forever, in principle, and the Law of Large Numbers wouldn't care, it's only making a promise about the center, not the shape.
Reading the chart
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1
Left edge: pure noise
At N = 1 the "average" is just one draw, as scattered as the population itself.
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2
The log x-axis is doing real work
Going from 10 to 100 trials narrows things about as much as going from 100 to 1000, equal ratios, not equal counts.
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3
Avg |error| tracks σ/√N
The same square-root shrinkage from the CLT page's standard error, just watched directly instead of inferred from a histogram.
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4
"New random sequences" changes the noise, not the destination
Every fresh set of six paths wiggles differently on the left and lands in the same place on the right.
This is the guarantee that makes averaging trustworthy at all: a single poll, a single A/B test, a single simulation run is one point on the far left of this chart, still mostly noise. Everything about "run more trials" or "collect more data" as advice is really just an appeal to slide this chart's N to the right.
The jargons
One promise: more draws, a tighter average, centered where it should be.
As the number of independent draws grows, their average converges to the population's true mean.
The average of all draws seen so far, recomputed after every new one, exactly what each line on the chart traces.
The population's actual average, computed exactly from its known distribution, the dashed target line.
How far a running average of N draws typically sits from the true mean. Shrinks with the square root of N, never hits exactly zero.
Each draw doesn't affect or depend on the others, the assumption underneath the whole guarantee.
The formal strength of the guarantee: convergence happens with probability 1, not just "usually" or "on average."