Sample Size Calculator

This calculator uses Cochran's formula, the standard method for sizing a survey that estimates a proportion. It answers a practical question: how many independent observations are needed so that the sampling error stays within the margin you specified?

The numerator grows with the variability of the answer, p(1 − p), and with the confidence you demand. The denominator shrinks as you tighten the margin of error, which is why halving the margin roughly quadruples the required sample. After you collect your data, you can run the observed result through our margin of error calculator to confirm the precision you actually achieved, or build the full range with the confidence interval calculator.

The formula regularly produces a fractional answer such as 384.16, but you cannot interview 0.16 of a person. Rounding to the nearest whole number would sometimes round down, and a sample of 384 does not quite deliver the promised 5% margin — the guarantee fails by a sliver. The calculator therefore always rounds up (a ceiling function), because the required size is a minimum: 385 is the smallest whole number of respondents that keeps the margin of error at or below the target.

This convention matters most when the raw result lands just above a whole number. A value of 384.02 still becomes 385, never 384. Treat the output as the floor for your recruitment plan, not a suggestion to aim near.

The term p(1 − p) reaches its maximum value of 0.25 exactly at p = 0.5, which is why 50% is the safe default: it produces the largest — most conservative — sample size, so your survey is protected no matter what the true proportion turns out to be. If a pilot study or last year's survey tells you the proportion is far from 50%, using that estimate shrinks the requirement. At 95% confidence with a 5% margin, an expected proportion of 20% needs only 246 respondents instead of 385.

Be careful with this shortcut: if your prior estimate is wrong and the true proportion is closer to 50% than you assumed, the achieved margin of error will be wider than planned. When in doubt, stay with 50%.

Cochran's formula assumes you are drawing from an effectively unlimited pool. When you sample a noticeable share of a small group — a company of 500 employees, a school of 2,000 students — each response covers a meaningful slice of the whole, and fewer responses are needed. The finite population correction (FPC) adjusts for this:

A common rule of thumb: the correction is worth applying when the sample would exceed about 5% of the population. For truly large populations — a city, a country — the adjustment changes almost nothing, which is why national polls of vastly different countries all use similar sample sizes.

A product team wants to estimate what share of customers would recommend their service, with a 5% margin of error at 95% confidence. They have no prior estimate, so they keep the expected proportion at 50%. The calculator works through these steps:

1. Square the critical value: z² = 1.96² = 3.8416.
2. Variability term: p(1 − p) = 0.5 × 0.5 = 0.25.
3. Square the margin: e² = 0.05² = 0.0025.
4. Apply the formula: n₀ = 3.8416 × 0.25 / 0.0025 = 0.9604 / 0.0025 = 384.16.
5. Round up: the required sample size is 385 respondents.

Now suppose the customer base is small: only N = 2,000 people. Applying the finite population correction: n = 384.16 / (1 + 383.16 / 2000) = 384.16 / 1.19158 ≈ 322.3955, which rounds up to 323 respondents — a saving of 62 interviews purely because the population is limited.

For comparison, keeping everything else fixed but changing one input at a time: 90% confidence needs 271 respondents, 99% confidence needs 664, and tightening the margin to 3% pushes the requirement to 1,068.