Sample Size for a Mean (Known σ)
How many samples you need to estimate a mean within a target margin of error, when the population standard deviation is known.
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The engineering
When you already know (or can confidently assume) the population standard deviation σ, the width of a confidence interval for the mean depends only on σ, the confidence level, and the sample size. Solve for n and you get the number of measurements needed so the interval half-width — the margin of error E — meets your target.
The n grows with the square of the ratio σ/E: halving the margin of error quadruples the sample count. A common gotcha is confusing E (the half-width of the interval) with the full interval width; enter half. Because n is always rounded up, the achieved margin row is slightly tighter than E — that's the safety you paid for with the ceiling.
Use this when σ is genuinely known from prior data or a spec; if you're estimating σ from the same sample, the t-distribution version (unknown σ) is the honest choice and requires iteration.
Where this math comes from
The sample-size formula falls straight out of the normal-theory confidence interval, which took shape as Jerzy Neyman formalized interval estimation in the early 1930s, building on Gosset's and Fisher's work on the distribution of sample means. Neyman's 1937 paper laid out confidence intervals as a repeated-sampling guarantee, and inverting the half-width to solve for n became standard planning practice for surveys and industrial sampling.
Douglas Montgomery's texts on statistical quality control and design of experiments carried this into everyday engineering practice, presenting n = (zσ/E)² as the first tool a quality engineer reaches for when scoping a measurement campaign with a known process standard deviation.
- 1908William Sealy Gosset ('Student')Publishes the distribution of the sample mean, groundwork for interval estimation.
- 1937Jerzy NeymanFormalizes the confidence interval as a repeated-sampling procedure.
- 1991Douglas C. MontgomeryCodifies n = (zσ/E)² as standard practice in Design and Analysis of Experiments.
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