How this calculator works
Enter your confidence level, margin of error, and expected proportion (use 50% if you have no prior estimate—it's the most conservative choice and produces the largest required sample size) to find how many survey responses or observations you need. If you know the total population size, a finite population correction reduces the required sample, since sampling a larger share of a small population reduces uncertainty faster.
A reverse mode is also available: enter the sample size you actually have (and the population, if known) to see the margin of error it achieves. This is useful when your sample size is fixed by budget or access, and you want to know how precise your results will be rather than how large a sample to aim for.
Cochran's formula
Uncorrected sample size: n0 = Z^2 * p(1-p) / e^2Finite population correction: n = n0 / (1 + (n0-1)/N)Z = 1.645 (90% confidence), 1.96 (95%), or 2.576 (99%); p = expected proportion; e = margin of error; N = population sizeWhen the population is unknown or very large, the correction has almost no effect and the uncorrected n0 is used directly.
Worked example: population of 1,000, 95% confidence, 5% margin of error
- Uncorrected: n0 = 1.96^2 * 0.5 * 0.5 / 0.05^2 = 3.8416 * 0.25 / 0.0025 = 384.16.
- Apply the finite population correction: n = 384.16 / (1 + (384.16-1)/1000) = 384.16 / 1.38316 = 277.74.
- Round up to a whole number of respondents: 278.
- This matches the widely published finite-population sample size tables for N=1,000 at 95% confidence and +/-5% precision.
Frequently asked questions
Why use 50% for the expected proportion if I don't know it?
The term p(1-p) in Cochran's formula is largest when p=0.5 (0.25), which produces the largest, most conservative required sample size. Any other assumed proportion would require fewer responses if your guess turns out to be right, but a wrong guess in the other direction would leave you under-sampled—50% protects against that.
What does the finite population correction actually do?
It reduces the required sample size when you're surveying a meaningful fraction of a small, known population. Sampling 300 out of a population of 1,000 already covers 30% of everyone, so less additional sampling is needed for the same confidence, compared with an unknown population of millions.
Where do the Z-scores 1.645, 1.96, and 2.576 come from?
They're the standard normal distribution's critical values for two-tailed confidence intervals: 90% confidence leaves 5% in each tail (Z=1.645), 95% leaves 2.5% in each tail (Z=1.96), and 99% leaves 0.5% in each tail (Z=2.576). These are the most commonly published values in statistics references.
What if my required sample size is larger than my population?
That means you'd need to survey more people than exist—in that case, the calculator caps the answer at a full census of the population, since you can't need more responses than the total number of people available.
Does a bigger sample always mean a smaller margin of error?
Yes, but with diminishing returns: because the formula involves a square root, quadrupling your sample size only halves your margin of error. Going from 100 to 400 responses cuts your margin in half, but going from 400 to 700 only helps a little further.