Why 50% is the safe assumption
The expected proportion feeds into the formula as p(1−p), which peaks at exactly p = 0.5. Assuming 50% therefore gives the largest — safest — sample size, and it is what you should use unless you have solid prior data. If you genuinely expect a lopsided split, say 10%, the required sample drops substantially, but guess wrong and your margin of error is bigger than advertised.
The population size barely matters
The single most counter-intuitive result in survey design: polling a city of 100,000 and a country of 100 million needs almost the same sample. The finite population correction only bites when your sample is a meaningful fraction of the whole group — surveying 200 people out of 500 employees, say. Leave the population blank for anything large.
Note too that this is the number of completed responses. If you expect a 20% response rate, you need to contact five times as many people.
Frequently asked questions
Why does halving my margin of error quadruple the sample?
Because precision improves with the square root of the sample size — the margin has n under a square root. Going from ±5% to ±2.5% costs four times the responses, and ±1% costs twenty-five times.
Can I rely on this for coursework or research?
Checked against the worked examples in standard texts. The assumed proportion changes the answer; 0.5 is the conservative default when you have no prior estimate.
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