Sample Size for a Mean
Worked example: 95% CI, sigma 15, E 2 → n = 216.09 — press Try an example to run it live, then adjust anything.
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Sample Size for a Mean explained
Every study design eventually reduces to this line: decide how much error you can live with, decide how confident you want to be, guess the standard deviation, and the sample size falls out. Estimating a mean to within 2 units at 95% confidence when σ ≈ 15 needs n = (1.96 × 15/2)² = 216.09 — always round up, so 217. Rounding down is the classic slip; it leaves you fractionally short of the precision you promised. The σ you plug in is usually borrowed from a pilot study, published literature, or the crude rule that a range spans about four standard deviations.
The square in the formula is the brutal part of research economics. Precision is bought at quadratic cost: tightening E from 2 to 1 in the example above takes the requirement from 217 to 865. Note too that the population size never appears — a well-drawn sample of 1,000 estimates a mean about as well in a city of 100,000 as in a country of 100 million, which is the single most counter-intuitive fact in survey work. Only when the sample is a sizeable fraction of a finite population (over about 5%) does a correction factor start to help you.
Sample Size for a Mean formula
- = Required sample size
- = Critical z-value
- = Standard deviation
- = Target margin of error
Missing one of these? Work it out first, then come back
- Required sample size — Sample Size for a Proportion, Standard Error of the Mean
- Critical z-value — Margin of Error for a Mean, Confidence Interval Lower Limit
- Standard deviation — Z-Score (Standard Score), Variance and Standard Deviation
- Target margin of error — Sample Size for a Proportion, Margin of Error for a Mean