Data needed to cut the standard error to a fifth

Question

A lab runs n = 12 assays to estimate a concentration. Management wants the standard error reduced to one-fifth of its current value. How many assays are needed in total?

Accepted Answer

The standard error of an average scales as 1/\sqrt{n}: 	ext{SE} = \frac{\sigma}{\sqrt{n}}. To shrink the SE by a factor, you must multiply the number of assays by that factor squared, since the count sits under the square root. Five times the precision (SE down to one-fifth) needs 5^2 = 25 times the data: n{	ext{new}} = 25 	imes 12 = 300. A 12-assay pilot balloons to 300 assays for a fivefold precision gain. Takeaway. SE falls like 1/\sqrt{n}: one-fifth the SE means 25	imes the assays. Each additional factor of precision gets dramatically more expensive.

Data needed to cut the standard error to a fifth

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