Two Proportions

= ( + )² ·

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Calculate

Example: a candidate polls at 38%, and the campaign wants to know whether an ad takes that number to 41%. How many people must be heard in each group, with and without the ad, to detect the difference with 80% power at 5% significance? The same goes for a conversion A/B test. Leave the design effect blank for simple random samples. Change the values to work out your own case.

Proportion in group 1 p₁ (%) Proportion in group 2 p₂ (%) Significance α (%) Power (%) Design effect deff

Step by step

What each term means

Size of Each Group (n): How many people to hear in each of the two equal-sized groups for the z-test to detect the difference with the requested power. The total is twice that.
Threshold under H₀: The critical value z_α/2 times the spread of the difference when H₀ holds and both groups have the average proportion p̄ = (p₁ + p₂)/2. For α = 5%, z_α/2 ≈ 1.96. It is how far the observed difference can go by chance.
Slack under H₁: The power quantile z_β times the spread of the difference when each group has its own proportion. For 80% power, z_β ≈ 0.84. It is the margin that makes the real difference clear the threshold in most samples.
Squared Difference: The difference you want to detect, as a proportion (0.03 for 3 points). Because it is squared, halving the difference quadruples n: detecting 3 points costs four times as much as detecting 6.
Design Effect (deff): How much the survey design inflates the variance compared with a simple random sample: clusters usually give 1.5 to 3, and strata can give less than 1. It multiplies n. Left blank, it equals 1. The poll simulator shows where it comes from.