A sample-size calculator returns the minimum number of subjects needed to detect an effect at a given confidence level and margin of error. The Toolenza calculator uses n = (z² × p × (1 − p)) / e² for proportions, where z is the confidence-level critical value and e is the margin of error.

Rules of thumb

For a survey with 95% confidence and ±5% margin of error, the worst-case sample size (p = 0.5, maximum variance) is:

• n ≈ 384 for any population large enough to ignore finite-population correction.
• n ≈ 1,067 if you want ±3% margin (the polling industry standard).
• n ≈ 9,604 for ±1% margin — why national polls almost never go that tight.

For an A/B test detecting a 10% relative improvement on a baseline 5% conversion rate with 80% power: you need about 15,000 visitors per arm.

When you need more (or fewer)

• Rare events — if you're measuring something that happens 1% of the time, n grows roughly inversely with p.
• Subgroup analysis — if you want to slice by age × gender × region, each cell needs its own n. The headline sample doesn't help.
• Small populations — if you're sampling 1,000 employees, finite-population correction shrinks n considerably (apply the correction factor (N − n) / (N − 1) under the square root).

The most common error

Running the math after the survey, not before. A survey of 50 people can't claim "results valid to ±5%" regardless of how it's framed. Sample size is a pre-registration decision — confirm before collecting data, not after.