Sample size calculation happens before data collection, and its purpose is narrow: to decide how many observations are needed so that, if an effect of a given size really exists, the study has a decent chance of detecting it. Two quantities drive the answer — the effect size you want to detect, and the error rates you tolerate. Smaller effects need more data, and the relationship is quadratic: halving the effect roughly quadruples the sample. α is the false-positive risk you accept, and power (1 − β) is the chance of finding a genuinely present effect, with 0.80 the customary floor. The means mode returns participants per group; the proportion mode returns a total.
Two means solves the two-sample t-test exactly. The normal approximation 2 (z₁₋α/tails + z_power)² / d² is only the starting estimate — it is optimistic, because it ignores the fact that the critical value comes from the t distribution and that the t distribution's degrees of freedom depend on n. The tool therefore evaluates the actual power of the test at each candidate n, using the noncentral t distribution, and returns the smallest group size whose achieved power reaches the target. The α term of the starting estimate is 1.960 for α = 0.05 two-sided, 2.576 for α = 0.01 two-sided or 1.645 for α = 0.05 one-sided; the power term is 0.842 for 80%, 1.282 for 90% and 1.645 for 95%. The critical t value actually used, with its df, and the power achieved at the returned n are both printed.
One proportion uses a different formula for the margin of error around an estimated percentage: n = z² p (1 − p) / e², with z = 1.960 at 95% confidence and 2.576 at 99%. Because p(1 − p) peaks at p = 0.50, an expected proportion of 50% gives the largest, safest sample.
With α = 0.05 two-sided and 80% power, an expected d of 0.20 needs 394 per group (788 in total); d = 0.50 needs 64 per group (128 total); d = 0.80 needs 26 per group (52 total); and d = 1.00 needs 17 per group. The error rates move the medium-effect case substantially: d = 0.50 at 80% power needs 64 per group, but a one-sided α = 0.05 needs 51, α = 0.01 two-sided needs 96, and 90% power needs 86. The proportion mode at 95% confidence gives 385 respondents for an expected 50% at ±5%, and 1068 at ±3%.
These are the figures R's power.t.test and G*Power return for the same specifications, which is the check worth making: sample-size tools that stop at the normal approximation are typically one to three participants light, and at α = 0.01 the gap is wide enough to matter.
The smallest effect that would matter, not the largest you hope for: if prior work suggests 0.50 but the smallest meaningful difference is 0.35, plan for 0.35.
They ignore the effect size, the main thing determining whether a study can detect anything; a fixed number works only if the effect is large.
Because there is only one group — the formula sizes a single survey estimate.
No — planning answers a question about the future. If the data are collected, run the appropriate analysis and report the effect size with its interval.