Sample Size Calculator

Frequently Asked Questions

Enter your margin of error and confidence level, pick the estimate type, and the calculator does the algebra: for a mean it computes n₀ = (zσ/E)², and for a proportion it computes n₀ = z²p(1−p)/E², then rounds up. If you enter a population size, the finite population correction lowers the requirement automatically. The reverse mode converts a sample size you already have back into the margin of error it delivers.

It depends on the decision the estimate feeds. National polls conventionally report ±3 percentage points, brand trackers commonly accept ±5, and safety-critical or census-adjacent work demands ±1. Halving the margin quadruples the required sample, so the right margin is the widest one your decision can tolerate – the benchmarks table on this page shows the full price grid.

It is the unrounded requirement for a proportion estimated at ±3 percentage points with 95% confidence and a worst-case 50% expected proportion: 1.96² × 0.25 / 0.03² = 1,067.11. Rounding up – the safe convention this calculator follows – gives 1,068. Published tables and methodology notes often quote the unrounded 1,067, which is why both numbers appear in the wild.

Cochran’s formula is the full z-based computation: n₀ = z²p(1−p)/E² for a proportion (or (zσ/E)² for a mean with known standard deviation), optionally followed by the finite population correction n = n₀/(1 + (n₀−1)/N). Published in W. G. Cochran’s 1977 textbook Sampling Techniques, it is the standard citation when every assumption must be stated explicitly. This calculator implements it directly.

Slovin’s formula is n = N/(1 + Ne²), where N is the population size and e is the error tolerance as a decimal. It is the compressed single-line version taught in introductory research methods, delivering roughly 95% confidence when e is taken as 5%. For 5,000 students at e = 0.05 it gives 370.37, rounded up to 371. This calculator reports the Slovin figure alongside the Cochran result whenever you enter a population size with a proportion estimate.

Raosoft is a long-running online sample size calculator that popularised the standard input set – 95% confidence, 5% margin of error, 50% response distribution – and it computes the same z-based mathematics this page implements. The 384/385 figure Raosoft returns for those defaults is the same number this calculator’s benchmarks table shows in its 95%/±5% cell.

Only when the sample is a meaningful share of the population. The finite population correction changes the requirement when n₀ exceeds roughly 5% of the population: surveying a 500-member loyalty programme drops the requirement from 70 to 62, while a national poll of 1,068 against millions of voters changes by less than one respondent. Leave the population field blank for large or unknown populations.

95% is the reporting standard across polling, research, and quality work, and it is this calculator’s default. Choose 90% when the cost of a wrong interval is low and every respondent is expensive; choose 99% when the decision is high-stakes, accepting that the jump from 95% to 99% inflates the sample by about 73%. State the level you used – an unstated confidence level makes the margin of error meaningless.

Because the proportion term p(1−p) reaches its maximum of 0.25 exactly at p = 0.5 and shrinks toward zero as p approaches 0 or 1. A 50% expectation is the worst case, so sizing on it guarantees the study is large enough whatever the true proportion turns out to be. When prior evidence says the proportion is near 8%, the honest requirement falls from 385 to 113 at ±5% and 95% confidence.

Statistically, a larger sample is always more precise, but practically the extra precision can be worthless. Going from ±3 to ±2 points costs 1,333 additional respondents, and beyond some point the interval describes differences too small for anyone to act on. Very large samples also make trivial differences statistically detectable, so report effect sizes alongside significance, and size to the decision rather than to the budget.

Run a pilot first. Even 20 observations produce a far better σ estimate than a guess, and our Standard Deviation Calculator computes it directly from raw values. Published values from similar studies are the next best source. Because the required size scales with σ², an underestimated standard deviation silently underpowers the study – when in doubt, inflate σ slightly and let the achieved margin absorb it.

Divide the required sample size by your expected response rate. If the calculation calls for 385 completes and history says 30% of contacted people respond, invite 385 / 0.30 ≈ 1,283 people. Non-response also biases results when non-respondents differ from respondents, which no sample size fixes – see the What Sample Size Cannot Fix section on this page.

Because respondents come in whole numbers and the formula does not. A computed size of 1067.11 means 1,067 responses deliver a margin of error a fraction over target; only the 1,068th response closes it. Rounding down silently under-delivers the precision you promised, so the convention – and this calculator – always ceils, and shows the achieved margin so you can see the slight over-delivery for yourself.

The two tools are companions. The Z-score Calculator converts between values, z-scores, percentiles, and p-values for data you already have; this calculator uses those same critical Z-values to plan data collection before it happens. A 95% confidence level here is a critical z of 1.96 there, and the z-score page’s critical value table is the source of every confidence level this page offers.

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