Practice question · Multiple choice
The standard error of the sample mean is σ/√n. Why the square root - why does quadrupling the sample only halve the uncertainty?
Hints
- Variances add for independent quantities; standard deviations do not. Which one is the natural quantity here?
- The variance of the sample mean is σ²/n. Now take the square root.
Show the answer
C. Because variance falls like 1/n, so its square root falls like 1/√n.
Why
Work in variances, where the arithmetic is clean: independent variances add to nσ², dividing by n to form the mean divides by n², leaving σ²/n, and the square root converts that back into the units of the data. Hence the diminishing return: halving uncertainty costs four times the data, which is why polls settle around a thousand respondents.
Practise Sampling and the Central Limit Theorem
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More questions on Sampling and the Central Limit Theorem
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