Sample size calculator for surveys and studies: n for a margin of error on a proportion or mean, and the per-group n for a target power.
Aktualisiert Geprüfte Beispiele: 6
A random sample of 385 gives a 95% confidence interval no wider than ±5 percentage points whatever the true proportion turns out to be (p = 50% is the worst case). Halving the margin roughly quadruples the sample; the formula assumes simple random sampling and says nothing about non-response or bias.
Always rounded up: rounding down would miss the target margin.
For a survey proportion the sample size is n₀ = z²p(1 − p)/E², where z is the critical value for the confidence level and E the margin of error; for a mean it is (zσ/E)². A known population size N reduces this through the finite population correction, n = n₀/(1 + (n₀ − 1)/N). To compare two means, each group needs 2(z₁₋α/₂ + z₁₋β)²/d², where d is the difference to detect divided by σ.
Survey planners and researchers use it before collecting data. The default, 95% confidence and ±5 points with p = 50%, needs 384.1 respondents, rounded up to 385; for a population of 10,000 the correction lowers that to 370.
Results are always rounded up. The formulas assume simple random sampling; cluster samples, weighting and non-response all call for a larger sample.
Prüfquelle: Cochran's formula 1.959964² × 0.25 / 0.05² = 384.146 (Python NormalDist); the widely published figure is 385
Prüfquelle: Finite population correction n₀/(1 + (n₀ − 1)/N) in Python
Prüfquelle: Finite population correction in Python: 384.146/(1 + 383.146/100)
Prüfquelle: (1.959964 × 15 / 3)² = 96.04 in Python
It is what Cochran's formula gives for 95% confidence, a ±5-point margin and p = 50%: 1.96² × 0.25 / 0.05² = 384.1, rounded up to 385. Because p = 50% is the worst case, 385 is enough for any true proportion in a large population. A population of 1,000 needs only 278, because each respondent is a larger share of it.
Only when the sample is a noticeable fraction of the population. The finite population correction turns the 385 needed for ±5 points at 95% into 370 for a population of 10,000 and 80 for a population of 100, while any population above 100,000 needs 383 to 385. A sample of 1,000 gives about ±3.1 points whether the population is 100,000 or 300 million.
Use 50% unless you have a reliable prior estimate. p(1 − p) is largest at p = 0.5, so that sample is big enough whatever the true value turns out to be. A prior estimate lowers the requirement: at 95% confidence and ±5 points, an expected 20% needs 246 respondents instead of 385. If the guess is wrong, the achieved margin will be wider than planned.
The margin shrinks with the square root of n, so halving it needs four times the sample. At 95% confidence and p = 50%, ±5 points needs 385 respondents, ±3 points 1,068, ±2 points 2,401 and ±1 point 9,604. Choosing 90% confidence instead of 95% lowers the ±5-point figure to 271.
Power is the probability that a test detects an effect of a stated size when that effect is real; 1 − power is the Type II error rate β. Cohen (1988) proposed 80% as a conventional minimum. Detecting a difference of 7.5 when σ = 15 (d = 0.5) with a two-sided test at α = 0.05 needs 63 per group for 80% power by the normal approximation; Cohen's t-based table gives 64.
Die Genauigkeit hängt von Ihren Eingaben und den Annahmen der Methode ab. Die Dezimalrechnung nutzt 50 signifikante Stellen, doch Schätzungen, numerische Verfahren und Quelldaten können ungenauer sein. Die angezeigte Rundung beseitigt diese Grenzen nicht. Anhand unabhängiger Quellen geprüfte Rechenbeispiele: 6. Beispielsweise wird „Survey at 95%, ±5%, p = 50% (defaults)“ anhand von Cochran's formula 1.959964² × 0.25 / 0.05² = 384.146 (Python NormalDist); the widely published figure is 385 geprüft.
NIST/SEMATECH e-Handbook of Statistical Methods, §7.2.2.2 Sample sizes required; Cochran, W. G. (1977). Sampling Techniques, 3rd ed., §4.4 (sample size for proportions and the finite population correction); Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences, 2nd ed., Table 2.4.1.
Dieser Rechner enthält 6 Rechenbeispiele mit Ergebnissen aus unabhängigen Quellen. Sie laufen in der Testsuite und können auch hier ausgeführt werden.
Confidence interval calculator for a mean (t or z), a proportion (Wilson or Wald) or a difference of two means, with the margin of error and both bounds.
t-test calculator for one-sample, two-sample (Welch or pooled) and paired tests, plus z-tests for a mean or proportion: p-value, critical value, decision.
Normal distribution calculator: z-scores, the probability below, above or between values, and the value at any percentile (inverse normal).
Chi-square test calculator for goodness of fit and independence in a contingency table: χ², p-value, expected counts and Cramér's V.
One-way ANOVA calculator: the F statistic, p-value, sums of squares, mean squares and η² for two or more groups, with the full ANOVA table.