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Sample Size Calculator

Find the minimum sample size for a survey or experiment: one-proportion studies with a margin of error, or two-group comparisons powered at 80% with α = 0.05.

Required sample size: 1,068

Required n per Group (Two Means, 80% Power, α = 0.05)

Effect size (Δ/σ)n per group
0.2 (small)393
0.5 (medium)64
0.8 (large)26
1.017

Values from n = 2·(1.96 + 0.842)²/δ² — the classic Cohen effect-size planning table.

The Formula Behind the Result

  1. Proportion mode: enter expected prevalence (or 0.5 for maximum conservatism) and your acceptable margin of error.
  2. The tool applies the standard normal-approximation formula with a finite population correction when you supply a population size.
  3. Means mode: enter the standard deviation and the smallest difference worth detecting to get the per-group n at 80% power.
  4. Both modes use two-sided α = 0.05; the comparison mode shows what higher or lower power does to the requirement.
Formula & Technical Standard
Proportion: n = z²·p(1−p)/e² (× N/(N+n−1) with finite population). Means: n per group = 2·(z_α/2 + z_β)²·σ²/Δ² with z_α/2 = 1.96, z_β(80%) = 0.842.

Worked Example

National survey
Input: p = 0.5, margin ±3%, population 1,000,000
Output: n = 1,067
The familiar "±3%" poll number requires about 1,067 respondents — the same regardless of whether the population is a city or a country, thanks to the correction.
A/B test on conversion
Input: baseline 5%, detect 5.5% (Δ = 0.5 pp), σ ≈ 0.218
Output: ≈ 3,762 per group at 80% power
Small lifts over small baselines are expensive: detecting a half-point improvement needs thousands per variant, which is why underpowered tests run "inconclusive".

Common Questions

Why does p = 0.5 maximize the required sample size?

The p(1−p) term peaks at 0.5. If you do not know the expected proportion, planning with 0.5 gives the largest, safest n — a survey sized for the worst case is valid for any result.

What does 80% power mean?

If the true effect equals your minimum detectable difference, an 80%-powered study finds a significant result 8 times out of 10. The remaining 20% are false negatives. Raising power to 90% costs roughly a third more subjects.

Does the calculator handle cluster sampling or attrition?

No. These formulas assume simple random samples of independent observations. For clustered designs inflate n by the design effect (1 + (m−1)·ICC), and add your expected dropout rate to the final figure.

Is the finite population correction always valid?

It applies when sampling without replacement and n is a meaningful share of N (rule of thumb: above 5%). Below that the correction is negligible and can be ignored.

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Confidence Interval Calculator (Proportion)

Build a confidence interval for a proportion from successes and trials — with the Wilson interval that stays accurate for small samples and 0% or 100% results.

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Confidence Interval Calculator (Mean)

Compute a confidence interval for a population mean from raw data or summary statistics, using the t-distribution — correct for the small samples the normal formula gets wrong.

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Odds Ratio Calculator

Compute the odds ratio from a 2×2 contingency table with 95% confidence interval and p-value — the standard measure of association for case-control studies.