Survey sample size calculator
Estimate completed responses for a population proportion under a simple random sampling design. At 95% confidence, an anticipated proportion of 50% and a margin of sampling error of ±5 percentage points, the large-population calculation is 384.16, rounded up to 385 completed responses. For a known population of 1,000 sampled without replacement, the corresponding finite-population estimate is 278.
This is a planning approximation for a proportion, not a guarantee of representativeness or a calculator for every research design. It does not account for nonresponse bias, clustering, weighting, measurement error or the power needed to detect a change.
278 completed responses
Finite-population planning estimate for 1,000 eligible units.
556 invitations at the assumed 50% usable-response rate.
Show the calculation
n₀ = 384.16; n = 384.16 / (1 + 383.16 / 1,000) = 277.74; round up to 278.
Assumes a simple random sample for estimating a proportion; the finite correction assumes sampling without replacement from the stated population. Invitations divided by an expected response rate are a field-planning estimate, not a correction for nonresponse bias.
How the sample-size formula works
For a large population, the usual normal-approximation planning formula for a proportion is n₀ = Z² × p × (1 − p) / e². Here, Z is the critical value for the chosen confidence level, p is the anticipated proportion and e is the desired margin expressed as a decimal.
For a known finite population of size N under simple random sampling without replacement, use n = n₀ / [1 + (n₀ − 1) / N]. Apply the correction to the unrounded n₀, then round the final result up to a whole completed response.
Penn State's sampling-methods lesson explains sample-size planning for proportions and the finite-population correction. The calculator uses the normal critical values 1.645, 1.960 and 2.576 for 90%, 95% and 99% confidence respectively.
Worked example: 1,000 eligible people
- Large-population calculation: 1.96² × 0.5 × 0.5 / 0.05² = 384.16.
- Finite correction: 384.16 / [1 + 383.16 / 1,000] ≈ 277.74.
- Round upward: plan for 278 completed responses under the assumptions.
- Invitation scenario: at an assumed 50% usable-response rate, 278 / 0.5 = 556 invitations.
At an assumed 25% usable-response rate, the same target implies 1,112 invitations, more than the 1,000 eligible people. That is a feasibility problem, not permission to count repeat reminders as additional sampled people.
Inviting the whole population may be reasonable, but nonresponse can still prevent a complete census and introduce bias. Adjust the collection plan and report what actually happened.
Sample-size table by population
These figures use the same approximation with p = 0.5, finite correction where N is known, and upward rounding. They are planning values, not universal validity thresholds.
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| Population | 90% confidence, ±5 points | 95% confidence, ±5 points | 95% confidence, ±3 points | 99% confidence, ±5 points |
|---|---|---|---|---|
| 50 | 43 | 45 | 48 | 47 |
| 100 | 74 | 80 | 92 | 88 |
| 250 | 131 | 152 | 203 | 182 |
| 500 | 176 | 218 | 341 | 286 |
| 1,000 | 214 | 278 | 517 | 400 |
| 2,500 | 245 | 334 | 749 | 525 |
| 5,000 | 257 | 357 | 880 | 586 |
| 10,000 | 264 | 370 | 965 | 623 |
| 50,000 | 270 | 382 | 1,045 | 655 |
| Large population | 271 | 385 | 1,068 | 664 |
For N = 500, the 95% / ±5-point estimate is approximately 217.49, so the upward-rounded target is 218, not 217. For the large-population setting, 384.16 rounds upward to 385, not 384. Keep the rounding rule consistent between text, table and calculator.
What confidence and margin of error mean
A 95% confidence procedure is designed so that, under its assumptions over repeated samples, about 95% of the intervals constructed contain the fixed population value. It does not guarantee that the value from this particular survey lies within five points of the estimate.
The margin describes sampling uncertainty under the design. It does not include all sources of error, such as leading questions, missing population coverage or systematic nonresponse. A precise estimate of a biased responding group can still be misleading.
For a large population, halving the margin approximately quadruples the sample requirement under this formula. Increasing confidence from 95% to 99% also raises the requirement. The finite correction changes the size of those effects when the sample is a substantial fraction of a known population.
Choose precision and confidence in relation to the decision and design. There is no universal rule that 95% / ±5 points is sufficient for every program, funder or high-stakes decision.
When the finite-population correction applies
The correction concerns the sampling design and fraction, not a hard cutoff such as “only populations under 5,000.” It applies to simple random sampling without replacement from the stated finite population. Its effect becomes smaller when the sample is a small fraction of that population.
Entering the size of an email list does not automatically justify the correction if that list does not cover the target population or responses come from an uncontrolled opt-in process. Define the population and frame before choosing N.
Sample size is not a validity certificate
A survey with 500 responses can have serious coverage, nonresponse or measurement problems. A smaller, appropriately designed study can still provide useful information with the uncertainty and scope clearly stated. There is no universal 30-response boundary below which all percentages become meaningless.
Review how people were selected, who could respond, who was missed and how questions were asked. AAPOR's transparency guidance distinguishes reporting for probability and nonprobability designs and asks researchers to disclose the methods supporting their precision claims.
An unrestricted open link does not become a simple random sample when it reaches the calculator's number. Do not attach this calculator's margin to such results without an appropriate, explicitly justified method.
When you need a different calculation
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| Your question or design | Why this calculator is not enough |
|---|---|
| Detect a pre/post change or difference between groups | Requires power planning with a meaningful effect, variability and the comparison design |
| Estimate a mean, such as an average score | Needs suitable information about variability and a mean-based formula |
| Clustered sample, such as learners within selected schools | Dependence within clusters affects precision |
| Weighted or complex sample | Design and weighting can change effective precision |
| Rare proportion or small expected counts | The normal approximation may be inadequate |
| Qualitative interviews or interpretive research | Sample adequacy depends on the method, scope and information sought, not this proportion formula |
NIST's treatment of sample sizes for testing proportions illustrates why detecting a change introduces additional planning quantities, including power. Do not use a one-proportion precision target as proof that a study can detect its intended effect.
There is also no universal range of 15–50 interviews that guarantees qualitative adequacy. For a mixed-method study, plan each strand according to its role and then plan how the evidence will be integrated.
Plan the breakdowns before fielding
An overall sample target does not guarantee useful precision for each location or demographic group. Decide which comparisons are essential and plan their sample sizes or precision appropriately. Avoid splitting a small dataset into many unstable or identifying groups.
If the design deliberately samples groups at different rates, the overall analysis may need weighting and appropriate variance estimation. Do not add subgroup results together without checking the design.
See equity metrics for denominators and interpretation, and survey analysis for reviewing the collected results.
Plan invitations using realistic assumptions
The simple planning equation is invitations ≈ ceiling(required usable responses / expected usable-response rate). Use an assumption grounded in comparable prior collection where possible, and test alternative scenarios.
Do not rely on unsourced channel benchmarks. An invitation sent to an active cohort, an external list and an unknown open-link audience are different situations. Record eligibility, partial responses, duplicates and other dispositions appropriately.
A response rate alone does not establish or remove nonresponse bias. AAPOR's standard definitions provide a framework for recording survey dispositions and outcome rates. The field plan should help the team understand who was reached and what usable evidence resulted.
For repeated surveys, plan matching and attrition separately
Decide whether the question concerns a proportion at one wave, within-person change or a comparison across groups. Those require different planning. For paired change, usable matched observations matter, not just the total responses at each wave.
Plan for attrition using suitable prior evidence or scenarios, but do not assume that a larger baseline automatically removes attrition bias. Retain missingness, relevant respondent context and a clear matching process where the design requires it.
Anonymous repeated samples remain useful for some group-level questions. Use pre- and post-surveys and impact survey questions to connect the design to appropriate wording and timing.
The calculator is one part of the collection plan
After planning the sample, the team still needs clear definitions, usable invitations, appropriate access, quality checks and an analysis method. A stable record can support authorized matching, but it does not correct sampling bias or guarantee accuracy.
Sopact's connected approach supports recurring collection and review with the relevant person, organization or group context. Evaluate how staff manage corrections, missing waves, question changes, qualitative evidence and reproducible counts across a complete cycle.
The Membership & Networks course covers practical shared collection planning. For reporting the resulting evidence, use the report-writing guide and report examples, stating the actual design and limitations.
Watch: collection with useful context
This introduction covers the operational collection workflow. It does not replace the statistical design needed for a particular survey.
Frequently asked questions
How many responses do I need at 95% confidence and a five-point margin?
Under the calculator's simple-random-sampling proportion assumptions with p = 0.5, plan for 385 for a large population, 278 for N = 1,000 or 218 for N = 500. These are upward-rounded planning values, not validity guarantees.
Why do some guides say 384?
The unrounded large-population calculation using Z = 1.96 is 384.16. This calculator rounds upward, giving 385. Consistent upward rounding avoids understating the calculated target.
Does the finite correction apply only below 5,000 people?
No. It depends on a finite population sampled without replacement and the sampling fraction. Its practical effect declines as the population becomes large relative to the sample.
Can I use this for an open online survey?
It does not make an opt-in sample a probability sample. Use an appropriate analysis and describe the recruitment and inference limits rather than attaching this margin automatically.
Does this calculate the sample needed to detect improvement?
No. Detecting a change or group difference requires a power calculation suited to the outcome and design. A one-proportion precision target does not answer that question.
What if the invitation estimate exceeds the population?
The assumed response rate and target are not feasible with that number of eligible units. Revisit collection, precision or design. Repeated reminders to the same person do not increase the eligible population.
