Averages settle down, slowly
Take independent observations with the same mean and variance , and let be their average. The law of large numbers says converges to as grows. This is why a casino wins in the long run, why a Monte Carlo average converges to the true expectation, and why estimating an average return from more data is better than from less.
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How fast? The average has the same mean but a variance of , so its typical error, the standard error, is . Accuracy improves only like the square root of the sample size: to halve the error you need four times the data, and to gain one more digit you need a hundred times as much. The same square-root law governs the error of Monte Carlo simulations.
How much can ten years tell us?A stock has annual volatility . Ten years of annual returns give a standard error of the mean of . A plausible true expected return is 6 to 8%, so the estimate's uncertainty is as big as the quantity itself: ten years cannot tell a 7% stock from a 0% one with any confidence. To pin the mean down to you would need years.
- LLN: the sample mean converges to the true mean as grows.
- Standard error: ; four times the data halves it.
- Not a law of compensation: past luck is not corrected by future luck.
The gambler's fallacyAfter five losing trades in a row, a win is not “”. Independent trials have no memory: the law of large numbers works because new data swamps the early run, not because the run is cancelled. The average of the first 1000 trials is dominated by later ones, yet the total profit can drift ever further from its expected value, by about .
Lab · what's graded
- A standard error15 pts
- The sample size you need15 pts
- What the law says15 pts
- What it does not promise15 pts
- Monte Carlo accuracy15 pts
- Ten years of data15 pts
- Why ten years is not enough10 pts