Why does halving your error need four times the data?

Question

You are estimating an unknown mean (say the average edge of a trade) from $n$ independent samples, and your error bar is currently too wide.

Why does the estimate's error shrink only like $1/\sqrt{n}$ , so that halving your uncertainty requires roughly four times as much data? And what kind of error does more data not fix?

Accepted Answer

The 1/\sqrt{n} law. The sample mean of n i.i.d. draws with standard deviation \sigma has standard error \operatorname{SE}(\bar{X}n) = \frac{\sigma}{\sqrt{n}}. This is the central limit theorem's rate: fluctuations of the average die like 1/\sqrt{n}. Because the error scales with the square root of n, cutting it in half means multiplying n by 4; a tenfold-tighter estimate costs 100 times the data. Precision gets expensive fast, which is why long track records and large samples are so prized, and why "just collect a bit more data" rarely rescues a marginal signal. What more data cannot fix: bia…

Why does halving your error need four times the data?

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