Math & Statistics
Sample Size Calculator
Estimate a proportion sample size with the standard large-population formula, apply finite-population correction, and translate the adjusted completion target into invitation counts at 70% and 50% response. The page exposes the assumptions that can make a real study require more observations.
Initial sample size-
Finite-population sample size-
Invitations at 70% response-
Invitations at 50% response-
Decision view
Sample requirement and response burden
Sample requirement and response burdenThe initial formula is corrected for the finite population, then translated into invitation counts at two explicit response-rate assumptions.
Exact scenario comparisonMargin of error (%) changes while all other entered assumptions remain constant.
| Margin of error (%) | Initial sample size | Finite-population sample size | Invitations at 70% response | Invitations at 50% response |
|---|
How to use Sample Size Calculator
- Define the population and outcome proportion, then enter a critical z value and margin of error on percentage scales.
- Review both the initial large-population requirement and the finite-population adjusted number of completed responses.
- Round completed responses and invitations upward, then add design-effect, eligibility, attrition, subgroup, and data-quality allowances outside this simple model.
Calculator guide
Understanding Sample Size Calculator
A sample-size calculation connects four design choices: the confidence multiplier, expected proportion, target margin of error, and population size. The result is a statistical requirement under a simple random-sampling model—not automatically the number of invitations, completed records, or usable subgroup responses a project will obtain.
Calculation method
How the calculation works
Apply the standard proportion sample-size formula and then use the finite-population correction for the entered population. Apply the standard proportion sample-size formula and then use the finite-population correction for the entered population.
Fieldwork plan
Bridge statistical sample to recruitment volume
The formula ends at completed responses; a usable recruitment plan must account for every stage that can remove cases.
Document every denominator so 'response rate' and 'completion rate' cannot be confused during fieldwork.
Worked situations
Practical examples
- With z = 1.96, expected proportion 50%, and margin of error 5%, the large-population requirement is approximately 384.146 completed responses.
- For a population of 100,000, finite-population correction changes that requirement only slightly because the sample is a small share of the population.
- If 70% of invited eligible people respond, divide the adjusted completion target by 0.70 and round up; the displayed invitation figure should never be rounded down.
Better inputs
Useful tips
- Use 50% when no credible planning proportion exists because p times one minus p is largest at 0.5 and therefore produces the most conservative requirement.
- Plan important subgroups separately; a sufficient overall sample can still leave a region, age group, or customer segment too small.
- Base invitation inflation on the eligible completion rate, not merely email delivery or initial contact rate.
Before relying on the result
Limitations and common mistakes
- The formula assumes a simple random-sample proportion and does not include clustering, stratification, unequal weights, repeated measures, or a design effect.
- The fixed 70% and 50% invitation references do not model ineligibility, partial completes, duplicate records, or differential nonresponse.
- Margin of error represents sampling precision only and does not correct coverage bias, nonresponse bias, measurement error, or poor questionnaire design.
Reference
Key terms
- Expected proportion
- Planning value p for the share expected to have the measured outcome.
- Margin of error
- Target half-width of the modeled confidence interval for the proportion.
- Finite-population correction
- Reduction applied when sampling without replacement from a population not much larger than the sample.
- Response rate
- Share of eligible invitations that produce usable completed responses under a clearly stated denominator.
Important note
Calculated directly from the entered values using the displayed formula and rounding settings.
Frequently asked questions
Why does 50% produce the largest sample?
For a binary proportion, p times one minus p reaches its maximum at 0.5, so uncertainty is largest there.
When does finite-population correction matter?
It becomes meaningful when the sample is not negligible relative to the population and sampling is without replacement from that defined population.
Should the displayed sample size be rounded up?
Yes. A fractional requirement represents more than the lower whole number, so completed responses and invitations should be rounded upward.
Does a larger sample eliminate bias?
No. More responses reduce modeled random sampling error but can leave systematic coverage, selection, nonresponse, and measurement bias unchanged.