Cochran's formula, from first principles
William Cochran's 1977 sampling text gives the sample size for estimating a proportion: n0 = z² p(1 − p) ÷ e². Every symbol earns its place. The z prices your confidence (1.96 for 95%), the e is the margin you will tolerate, and p(1 − p) is the variance of a proportion, at its worst when p = 0.5. The standard classroom numbers: 95% confidence, p = 0.5, e = 0.05 gives 385 (384.16 before rounding up), the sample size behind half the surveys ever fielded.
A city of N = 2,000 households. Cochran's own correction then applies, because 385 is nearly a fifth of the population: n = n0 ÷ (1 + (n0 − 1)/N) = 384.16 ÷ 1.1916 = 322.4, rounded up to 323 households. Sampling without replacement from a small list is more efficient than the infinite-population formula assumes, and the correction returns the savings: 62 fewer interviews for the same precision.
Choosing p honestly
The p you feed the formula should be a prior estimate of the proportion you are measuring: a pilot study, last year's survey, a registry figure. Using 0.5 when you actually expect 0.1 oversizes the study by a factor of about 2.8 (p(1−p) falls from 0.25 to 0.09). Using a hopeful p when the truth is nearer 0.5 undersizes it, and the margin you promised will not be met. When nothing is known, 0.5 is the defensible choice precisely because it cannot undersize; say so in the methods note.
Forgetting the response rate. Cochran sizes completed interviews. If 30% of invitees typically respond, the fieldwork list needs 385 ÷ 0.7 = 550 names. Sizing invitations with the completions formula is the commonest sampling arithmetic error in practice.
Frequently asked questions
What p should I use if I have no idea of the proportion?
0.5. It maximises p(1 - p), so the sample is sized for the worst case and cannot come up short on variance grounds.
How does Cochran differ from Slovin?
Cochran makes the confidence level (z) and proportion (p) explicit choices; Slovin's formula is the special case z = 2, p = 0.5 with the population correction built in, and never tells you.
When should the finite population adjustment be applied?
When the uncorrected sample exceeds roughly 5% of the population. Below that, the correction changes little; the calculator shows both numbers so you can see the size of the effect.