How much data to halve the standard error?

Question

You average n = 100 independent measurements, each with standard deviation \sigma = 20 (so variance \sigma^2 = 400), and form the sample mean \bar{X}. What is the standard error of \bar{X}? How many measurements would you need to cut that standard error in half?

Accepted Answer

Standard error now. For n independent draws, \operatorname{Var}(\bar{X}) = \sigma^2/n, so the standard error is 	ext{SE} = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{100}} = \frac{20}{10} = 2. Halving it. You want 	ext{SE} = 1, so \frac{20}{\sqrt{n}} = 1 \implies \sqrt{n} = 20 \implies n = 400. That is 4	imes the data (from 100 to 400) to halve the standard error, because SE falls like 1/\sqrt{n}: cutting it in half needs quadrupling the sample. Takeaway. SE = \sigma/\sqrt{n}, so precision improves with the square root of sample size; halving the standard error costs 4	imes the data, and o…

How much data to halve the standard error?

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