Draw real samples, watch the running mean settle onto the population mean, and meet the one famous distribution where it never settles at all.
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The Statement
The running mean of i.i.d. draws converges to the population mean
Take independent draws from one fixed distribution and keep updating the average as each draw arrives. The law of large numbers says this running mean homes in on the expected value of a single draw as n grows. That is the entire bridge between probability and data: relative frequencies earn the right to be called probabilities.
Fine print, kept light: the weak law says the chance of a noticeable gap between the running mean and the true mean shrinks to zero (convergence in probability), while the strong law says the sample path itself settles onto the true mean with probability one (almost surely). On screen they look the same: the line settles. Both require a finite population mean, and that clause matters below.
Sample It Yourself
Pick a distribution (see the data distributions guide for their shapes), then append real seeded draws. Start extra runs to see how different sample paths wobble differently yet converge to the same target.
Uniform on {1, 2, 3, 4, 5, 6}. Population mean is the face average.
Active run draws (n)
0
Running mean
no draws yet
Gap to true mean
no draws yet
Common Misreading
The Gambler's Fallacy
LLN says the proportion of heads converges. It does not say past deficits get corrected, and it does not say the raw counts balance out.
Two Theorems, One Sample Mean
LLN vs CLT
Law of Large Numbers
Answers WHERE the sample mean goes: to the population mean. It is a statement about the destination of the running average, and it says nothing about the shape of the wobble along the way.
Central Limit Theorem
Answers what SHAPE the fluctuations around that target take: rescaled by the square root of n, the sample mean's deviations from the population mean look approximately normal (given a finite variance).
In the chart above, LLN is the line settling onto the dashed target. CLT describes the statistics of the shrinking band it wiggles inside. Both need finite moments, which is exactly what the Cauchy panel takes away. See the fluctuations for yourself in the Central Limit Theorem guide.