Calculator Form
Plotly Graph
The graph shows how your recommended invitations change as the margin of error becomes tighter or wider.
Example Data Table
| Scenario | Confidence | Margin | Expected Proportion | Population | Design Effect | Response Rate | Completed Responses | Invitations |
|---|---|---|---|---|---|---|---|---|
| National opinion poll | 95% | 5% | Worst case | Large / Unknown | 1.00 | 100% | 385 | 385 |
| Customer satisfaction study | 95% | 4% | 60% | 10,000 | 1.20 | 80% | 654 | 818 |
| Safety compliance audit | 99% | 3% | 20% | 2,500 | 1.50 | 75% | 1,347 | 1,796 |
| Community health survey | 90% | 6% | 35% | 1,200 | 1.10 | 85% | 165 | 194 |
Formula Used
Base proportion sample size
n₀ = (Z² × p × (1 − p)) / E²
Here, Z is the z-score for the chosen confidence level, p is the expected proportion, and E is the margin of error in decimal form.
Finite population correction
n_fpc = n₀ / (1 + ((n₀ − 1) / N))
Use this when your target population is known and not very large. It reduces the required number of completed responses.
Design effect adjustment
n_design = n_fpc × DEFF
Clustered, weighted, or complex samples often need extra responses. A design effect above 1 increases the sample size.
Response rate inflation
n_invited = n_design / RR
RR is the expected response rate in decimal form. This converts completed-response targets into invitations needed.
How to Use This Calculator
- Enter a study label to identify your project in the export files.
- Choose the confidence level that matches your reporting standard.
- Enter the margin of error you can tolerate for the estimated proportion.
- Use an expected proportion from prior data, or choose the worst-case 50% option.
- Provide the target population only when it is known and limited.
- Increase the design effect for clustered or otherwise complex sampling designs.
- Enter the response rate you realistically expect from invitations sent.
- Submit the form, review the result card, then export CSV or PDF as needed.
Frequently Asked Questions
1) What does this calculator estimate?
It estimates how many completed responses you need to measure a population proportion within a chosen confidence level and margin of error. It also estimates invitations needed after accounting for nonresponse.
2) Why is 50% called the worst-case proportion?
A proportion of 50% produces the largest variance for a binary outcome. That variance creates the largest required sample size, making it a cautious planning choice when no prior estimate exists.
3) When should finite population correction be used?
Use it when your population size is known and relatively limited. If your required sample is a noticeable share of the full population, correction reduces the completed responses needed.
4) What is a design effect?
Design effect measures how much a complex sample increases variance compared with simple random sampling. Clustered or weighted designs often need larger samples, so a value above one is common.
5) Why does a smaller margin of error need more responses?
Tighter precision means the estimate must vary less across repeated samples. To reduce uncertainty, the study needs more completed responses, sometimes much more as the margin becomes very small.
6) Should I round sample size up or down?
Always round up for planning. Rounding down can leave the study underpowered for its precision target. This calculator reports rounded recommendations to support practical field decisions.
7) Does response rate change the number of completed responses?
No. Response rate affects how many invitations you should send, not how many completed responses the analysis requires. Lower response rates increase the outreach needed to hit the same completion target.
8) Can I use this for rare events or medical prevalence?
Yes, but choose the expected proportion carefully. Rare outcomes may still need sizable samples for narrow precision. For regulated or high-stakes studies, confirm assumptions with a qualified statistician.