Data needed to cut the standard error to a quarter

Question

A pilot survey of n = 20 people gives a rough estimate of an average. You want the standard error to be one-quarter of its current size. How many people must you survey in total?

Accepted Answer

Standard error shrinks like 1/\sqrt{n}: 	ext{SE} = \frac{\sigma}{\sqrt{n}}. To divide the SE by a factor, multiply the sample size by that factor squared, because the sample size lives under the square root. Making the estimate four times as precise (SE down to one-quarter) requires 4^2 = 16 times the data: n{	ext{new}} = 16 	imes 20 = 320. A pilot of 20 becomes a full study of 320 just to gain a factor of four in precision. Takeaway. SE \propto 1/\sqrt{n}: quartering the SE takes 16	imes the sample. Precision is expensive because it scales with the square root of effort.

Data needed to cut the standard error to a quarter

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