How many samples to halve the standard error

Question

An estimate is an average of independent, identically distributed samples. Right now you have 50 samples, giving a standard error of $\sigma/\sqrt{50}$ .

How many samples in total do you need so that the standard error is half of its current value?

Accepted Answer

The standard error of a mean is \sigma/\sqrt{n}, so it shrinks in proportion to 1/\sqrt{n}. To cut the standard error in half, you need \sqrt{n} to double, which means n must grow by a factor of 2^2 = 4. Starting from 50 samples: n{	ext{new}} = 4 	imes 50 = 200. This diminishing return is why precision is expensive: each further halving of the error costs four times the data. Takeaway. Because standard error scales as 1/\sqrt{n}, halving it needs 4x the samples; going from 50 to 200.

How many samples to halve the standard error

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