- Required sample size
- 385
- Critical z value
- 1.96
How it works
Complementary proportions need equal sample sizes
Expected proportions of 20% and 80% produce the same p(1−p) and therefore the same sample-size estimate when other settings match. The uncertainty depends on the balance between outcomes, not which outcome you call a success.
Planning precision before collecting data
With 95% confidence, a 50% expected proportion, a 5-percentage-point margin of error, and unlimited population, the normal approximation requires 385 observations after rounding up. Halving the margin to 2.5 percentage points increases the requirement to 1,537, roughly four times as many. The 50% proportion gives the largest p(1−p) when no better estimate is available. This count assumes random independent sampling; it does not correct selection bias or include extra invitations needed for nonresponse.
Required sample size: formula and assumptions
n₀ = z²p(1−p)/e²; finite correction n = n₀/[1+(n₀−1)/N].
Round up the required sample size. Random independent sampling is assumed; cluster designs and nonresponse may require larger samples.
Find margin of error: formula and assumptions
Margin=z√[p(1−p)/n], with finite-population correction √[(N−n)/(N−1)].
Assumes a random independent sample and the normal proportion approximation.
Worked example
Using the Required sample size example:
- Confidence level (%)
- 95
- Margin of error (percentage points)
- 5
- Expected proportion (%)
- 50
- Population size (0 = unlimited)
- 0
n₀ = z²p(1−p)/e²; finite correction n = n₀/[1+(n₀−1)/N].
Example result
Required sample size: 385
Critical z value: 1.96
These are illustrative inputs. Change the values to explore a different scenario.
Formula references
- OpenStax: Algebra and Trigonometry (opens in a new tab)
- NIST statistical methods handbook (opens in a new tab)
References explain the underlying methods; they do not imply endorsement or individual professional review of Finflo.
Common questions
Why is the required sample size rounded up?
A fraction of a participant cannot be sampled. Rounding up preserves the minimum count required by the stated approximation.
Does the calculation account for nonresponse or clustered sampling?
No. It assumes random independent sampling. Nonresponse and complex survey designs may require a larger planned sample.
Prepared by Finflo. See our methodology and editorial policy.