Free Research Tool
How many responses do you need?
Estimate the number of completed responses needed to measure a population proportion at a chosen confidence level and margin of error. You can also work the calculation in reverse and estimate the margin of error a completed sample already delivers.
Your result
Recommended completed responses
370
For a population of 10,000, 95% confidence, ±5% margin of error and a 50% expected proportion, the recommended minimum is 370 completed responses.
Response planning
Completed responses are the answers you need back. Invitations are the people you contact to get them.
- Completed responses required
- 370
- Expected response rate
- 100%
- Invitations to send
- 370
| Response rate | Invitations to send | Status |
|---|---|---|
| 25% | 1,480 | Workable |
| 35% | 1,058 | Workable |
| 50% | 740 | Workable |
| 75% | 494 | Workable |
- Population
- 10,000
- Confidence level
- 95%
- Margin of error
- ±5%
- Expected proportion
- 50%
- Design effect
- 1.00
- Response rate
- 100%
- Finite population correction
- Applied
Why this number?
We start by asking how precise the estimate needs to be. A ±5% margin of error at 95% confidence means that if the study were repeated many times, about 95% of the intervals would contain the true population value.
Because the expected proportion is set at 50%, the calculation assumes that level of variation in the answers. At 50% the variation is at its maximum, so the required sample is largest.
The sample size calculation runs in this order:
- Simple random sampling target: n₀ = z² × p × (1 − p) / e² = 384.15.
- Finite population correction: n = n₀ / (1 + (n₀ − 1) / N) = 369.97.
- Design effect: the corrected figure is multiplied by 1.00 = 369.97.
- Capped at the population size where one is known, then rounded upward.
The design effect is a conservative planning adjustment, not an exact correction for every complex design. Formal cluster, multistage, stratified, unequal-weight or repeated-measures studies can require design-specific calculations.
The expected response rate only estimates how many people to approach. It never changes the number of completed responses required, and the contacts figure is capped at the population size when one is known. The benchmark table shows the same target under other plausible response rates and marks the rates that cannot deliver it from a known population.
Sample size mathematics does not guarantee representativeness. Non-probability sampling, nonresponse bias, clustering, weighting and complex designs can all require specialist planning beyond this calculator.
z = 1.9600. Expected counts at the final sample size: 185.0 in the outcome category and 185.0 in the other category. Both should be at least 10 for the normal approximation to be dependable.
How the calculation works
This calculator works with a population proportion under simple random sampling, or a sampling design that is reasonably representative of the population. Sample size mode estimates the responses needed for a target margin of error. Precision mode does the reverse and estimates the margin of error a completed sample delivers. Both modes use the same assumptions, so a scenario gives consistent answers in either direction.
An expected proportion of 50% is used by default. When the true proportion is unknown, 50% is the conservative choice because p(1 − p) is at its maximum there and therefore produces the largest required sample for a given confidence level and margin of error.
A finite population correction reduces the required sample when you are sampling without replacement from a known finite population. For a large or unknown population, no correction is applied.
The design effect is 1 under simple random sampling and can be increased as a planning adjustment for clustered or otherwise complex designs. Formal stratified, multistage, cluster, unequal-weight or repeated-measures studies may require design-specific calculations.
The expected response rate estimates how many people may need to be approached. It does not change the target number of completed responses.
Sample size mode
n₀ = z² × p × (1 − p) / e²
n = n₀ / (1 + (n₀ − 1) / N)
The result is inflated by the design effect, capped at the population size where one is known, and rounded upward.
Precision mode
n_srs = n / DEFF
n₀ = n_srs × (N − 1) / (N − n_srs)
e = z × √(p × (1 − p) / n₀)
Precision mode is the algebraic inverse of the same formulas. It removes the design effect, undoes the finite population correction when the population is known, and reports the resulting margin of error. If the completed sample covers the whole population it is treated as a census and no sampling margin of error is reported.
When this calculator is not enough
This tool covers one common question. Separate calculations are required for other study designs, including:
- Estimating a mean rather than a proportion
- Comparing two groups
- Hypothesis testing and statistical power calculations
- Prevalence studies with special assumptions
- Cluster or multistage sampling
- Stratified designs with allocation rules
- Longitudinal or repeated-measures studies
- Very rare outcomes
Planning a more complex study?
Deep Dive Data can help with sampling methodology, questionnaire design, fieldwork planning, analysis and reporting.
Discuss Your ResearchReferences and methodology
The formulas used here are the standard normal-approximation sample size calculation for a proportion and the finite population correction, as described in the NIST/SEMATECH e-Handbook of Statistical Methods and in the population survey guidance behind CDC Epi Info StatCalc.
The result assumes the sample is random or otherwise representative of the target population. A large sample does not correct selection bias or poor survey design.