Calculating sample size...
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Calculate to see the minimum completed responses, response-rate-adjusted invitations, finite population correction, and full working.
What Is Sample Size?
Sample size is the number of completed observations used to estimate something about a larger population. For a survey proportion, that might be the share of customers who are satisfied, voters who support an option, or students who prefer a course format. A sample cannot remove every source of uncertainty, but a well-chosen target controls the amount of random sampling error expected under the model.
This sample size calculator is designed for one population proportion. It answers a planning question: how many completed responses are needed to estimate a percentage at a chosen confidence level and margin of error? It does not calculate statistical power for experiments or the sample needed to estimate a population mean.
How to Calculate Survey Sample Size
The unlimited-population formula is n₀ = z*² × p(1-p) / e². The critical value z* comes from the selected confidence level, p is the expected population proportion, and e is the margin of error written as a decimal. For 95% confidence, z* is approximately 1.96. A 5% margin is entered as 0.05 in the formula.
Suppose no prior estimate is available, so p = 0.50. With 95% confidence and a ±5% margin, n₀ is 384.1459. A sample target must be rounded up, giving 385 completed responses. Rounding down would no longer meet the requested precision.
Confidence Level, Margin of Error, and Proportion
Confidence level describes the long-run performance of the interval method. If the same random sampling process were repeated, a 95% method would produce intervals containing the true population proportion about 95% of the time. Raising confidence from 95% to 99% requires a larger sample because the interval must cover a wider range of plausible sampling outcomes.
Margin of error controls precision. Smaller margins require substantially more responses because e is squared in the denominator. The expected proportion also matters through p(1-p). That term is largest at 50%, so 50% gives the most conservative sample size when the likely result is unknown. A credible estimate from prior research may reduce the target, but an optimistic guess can leave a study underpowered for its intended precision.
Sample Size at 95% Confidence and 50% Proportion
Unlimited population, estimated proportion 50%. Bar lengths use a logarithmic scale so every value remains readable.
| Margin of error | Minimum sample |
|---|---|
| ±10% | 97 |
| ±5% | 385 |
| ±3% | 1,068 |
| ±2% | 2,401 |
| ±1% | 9,604 |
Halving the margin of error requires roughly four times the sample size when confidence and the expected proportion stay fixed.
Finite Population Correction
When sampling without replacement from a known, finite population, surveying a large share of that population gives more information than the unlimited-population formula assumes. The finite population correction is n = N × n₀ / (N + n₀ - 1), where N is the population size. For a population of 1,000, the standard 95% and ±5% target falls from 385 to 278 completed responses.
The correction matters most when the sample is a noticeable fraction of the population. It changes very little for a population of hundreds of thousands because a few hundred responses remain a tiny fraction. Leave population size blank when it is unknown or effectively very large.
How Response Rate Affects Invitations
The statistical formula returns completed responses, not invitations. If 278 responses are needed and the expected response rate is 60%, divide 278 by 0.60 and round up: invite about 464 people. The response-rate field changes only the outreach plan; it does not lower the number of usable responses required.
Response rate should be based on comparable surveys, audience engagement, contact method, reminders, and survey length. If the calculated invitations exceed a finite population, the calculator caps its recommendation at the population and warns that the assumed response rate may not produce enough completions.
Sample Size Reference Table
The chart above shows why sample-size decisions should be made before data collection. At 95% confidence and p = 50%, ±10% needs only 97 responses, while ±1% needs 9,604. More precision can be expensive, and collecting a very large sample is not automatically better if sampling methods or survey questions introduce systematic bias.
Assumptions and Limitations
The formula assumes a simple random sample, independent selections, a proportion outcome, and a normal approximation. Cluster samples, stratified designs, repeated observations, weighting, and complex survey designs may need a design effect or specialist analysis. For an experiment or A/B test, sample size should instead account for statistical power, significance level, baseline rate, and minimum detectable effect.
A large sample reduces random sampling error but does not fix undercoverage, nonresponse bias, leading questions, duplicate responses, or measurement error. The target should therefore be paired with a defensible sampling frame and collection plan. After collecting data, use the confidence interval calculator to quantify uncertainty around an observed result. The z-score calculator explains the critical values behind confidence levels, while the probability calculator supports related event calculations.
Sample Size Calculator FAQ
What sample size do I need for a 95% confidence level?
At 95% confidence, a 5% margin of error, and a conservative 50% estimated proportion, an unlimited population requires 385 completed responses. A known finite population may require fewer after correction.
Why is 50% used for the estimated proportion?
The expression p(1-p) reaches its maximum at 50%, producing the largest and most conservative sample target. Use it when there is no reliable prior estimate.
Does population size affect sample size?
Yes, when the required sample is a substantial part of a known finite population. For very large populations, the correction is small and the unlimited-population result is usually sufficient.
Should sample size be rounded up or down?
Always round up. The calculated value is a minimum, so rounding down would miss the requested confidence and margin target.
How do I account for a low survey response rate?
Divide the required completed responses by the expected response rate and round up. For example, 300 completions at a 50% response rate require about 600 invitations.
Does a larger sample remove survey bias?
No. A larger sample reduces random sampling error, but it cannot repair a biased sampling frame, poor questions, nonresponse bias, or invalid measurements.
Can I use this calculator for experiments or A/B tests?
No. Experiments and A/B tests require power analysis using an expected effect size, baseline rate, significance level, power target, and group allocation.
External Resources
Authoritative explanations of proportion sample-size planning and statistical assumptions.
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