Data needed to cut the standard error to a third

Question

A sensor currently averages n = 30 readings to estimate a temperature. You want to shrink the standard error to one-third of its current value. How many total readings do you need?

Accepted Answer

The standard error falls off as 1/\sqrt{n}: 	ext{SE} = \frac{\sigma}{\sqrt{n}}. Because n sits under a square root, driving the SE down by a factor requires multiplying the sample by that factor squared. To make the average three times as precise (SE down to one-third), you need 3^2 = 9 times the data: n{	ext{new}} = 9 	imes 30 = 270. So going from decent to three-times-sharper is not a small top-up, it is a ninefold increase in readings. Takeaway. Cutting the SE to 1/k costs k^2 times the data: one-third the SE means nine times the readings.

Data needed to cut the standard error to a third

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