Quick answer: The survey sample size formula is n = z² p(1−p) ÷ e². At 95% confidence (z = 1.96), a 5% margin of error, and p = 0.5, you need 385 completed responses — for almost any population size. For a 3% margin you need about 1,068; for 2%, about 2,401. Small known populations need fewer (finite population correction).

🔑 Key Takeaways

• 1385 is the magic survey number. At 95% confidence with a 5% margin of error, 385 responses is enough whether your population is 10 thousand or 10 million.
• 2Population size barely matters once it is large. Required n levels off fast — the finite population correction only helps when you sample a noticeable fraction of a small population.
• 3Precision is expensive: n grows with 1/e². Halving the margin of error (5% → 2.5%) roughly quadruples the sample you need.
• 4Use p = 0.5 when the proportion is unknown. It maximizes p(1−p) and therefore gives the safest, most conservative sample size.
• 5Comparing groups? Use power analysis, not the survey formula. n per group = 2(z_α/2 + z_power)² / d² — about 63 per group for a medium effect (d = 0.5) at 80% power.
• 6Plan for non-response. The formulas give completed responses; divide by your expected response rate (e.g., 385 ÷ 0.40 ≈ 963 invitations at a 40% response rate).
• 7Slovin's formula is a quick shortcut, not a replacement. n = N/(1+Ne²) silently assumes 95% confidence and p = 0.5 — fine for coursework, but report Cochran's formula in serious research.

🧮 Calculate Your Sample Size

📊 Results

Full Results Table

Visualizations

🛠 How to Use This Sample Size Calculator

1. Pick the goal. Survey / Proportion for polls and questionnaires, Estimating a Mean for measurements, or Comparing Two Groups for experiments and clinical trials — or load one of the ten sample scenarios.
2. Set the confidence level. 95% is standard; choosing 99% grows the sample, 90% shrinks it.
3. Set the margin of error. 5% is typical for surveys. Remember the inverse-square rule: halving the margin quadruples the sample.
4. Add the population size if it is small and known. The finite population correction can substantially cut the required sample for populations under a few thousand.
5. Click "Calculate Sample Size." You get the minimum n, the Slovin comparison, invitations needed after non-response, and four charts of the trade-offs — then copy a methods-section sentence from the reporting templates below.

📑 Sample Size Table: Common Survey Scenarios at a Glance

These are the numbers everyone looks up, computed with the Cochran formula (p = 0.5) and the finite population correction. Rows are population sizes; columns are margins of error at 95% confidence.

Notice how each column flattens as N grows — that is why "how big is the population?" matters far less than "how precise do you need to be?". The z-values behind this table are 1.645 (90%), 1.960 (95%), and 2.576 (99%).

ƒ Sample Size Formulas Used

🔍 Interpreting Your Sample Size — In Detail

The number this calculator returns is the minimum number of completed, usable responses or measurements needed to hit your stated precision or power. Three trade-offs govern it, and understanding them lets you negotiate with reviewers, clients, and budgets.

The three levers

Margin of error is the dominant lever. Because n scales with 1/e², tightening the margin from ±5% to ±3% takes you from 385 to 1,068 responses, and ±2% demands 2,401. If a survey budget is fixed, relax the margin rather than the confidence level. Confidence level is the second lever: moving from 95% to 99% multiplies the sample by (2.576/1.960)² ≈ 1.73. Population size is the weakest lever: it only helps when the population is small — at N = 500 a 5%-margin survey needs just 218 respondents instead of 385, but past N ≈ 20,000 the correction is negligible.

Reading the power analysis result

For two-group comparisons, the output is per group, so double it for the total. The driving force is the effect size you want to detect: small effects are brutally expensive (d = 0.2 needs ~394 per group at 80% power) while large effects are cheap (d = 0.8 needs ~26). If your computed sample is unaffordable, your realistic options are accepting detection of only larger effects, raising α (rarely acceptable), lowering power (risky — 80% is already the conventional floor), or using a stronger design (paired measurements, ANCOVA with covariates, or more precise instruments).

What the number does not fix

Sample size buys precision, not validity. A perfectly sized sample drawn from a biased frame (only daytime landline answerers, only volunteers) still misestimates the population — non-random selection error does not shrink with n. Likewise the formulas assume simple random sampling; cluster and stratified designs need a design-effect multiplier (commonly 1.5–2.0 for cluster samples). And remember the result counts completes: always inflate by your expected response rate before sending invitations.

✍ How to Write Sample Size in Your Methods Section

Journals and thesis committees expect an explicit sample size justification. Below are copy-ready templates — replace the bracketed values with your numbers from the results table above.

Template 1 — Survey (Cochran + FPC)

Template 2 — Slovin's formula

Template 3 — Estimating a mean

Template 4 — Power analysis (two groups)

Reporting checklist

State the formula or software used; report every input (confidence level or α, margin of error or effect size, power, assumed proportion or SD, population size); justify the effect size or SD with a citation or pilot data; describe the rounding and any inflation for non-response or attrition; and name the sampling method, since the formulas assume simple random sampling.

🔬 Example Results

Six worked examples matching the sample scenarios in the dropdown above. Load any scenario and click Calculate to reproduce the numbers exactly.

🧭 When to Use This Tool

Use this sample size calculator whenever you are planning a survey or questionnaire and need to know how many responses make it representative, designing an experiment or clinical trial and must justify participants per group, estimating a population mean to a stated precision, writing the sample-size paragraph of a thesis or grant proposal, checking a published study's power after the fact, or teaching sample size determination, Slovin's formula, or power analysis. It fits researchers, students, market researchers, UX teams sizing A/B tests, and anyone who must defend the question "why n = 385?"

⚠ Assumptions & Limitations

All formulas here assume simple random sampling; stratified designs need slightly fewer and cluster designs need more (multiply by a design effect, typically 1.5–2.0 for cluster surveys). The proportion formula uses the normal approximation, which weakens when n×p or n×(1−p) is below about 10 — for very rare outcomes use exact binomial methods. The mean formula requires an estimate of σ; if it comes from a small pilot, treat the result as a floor and add a safety margin. Power calculations depend entirely on the effect size you assume — an optimistic d produces an underpowered study, which is the single most common design failure in published research (Button et al., 2013). The outputs count completed, usable observations: inflate for non-response, attrition, and invalid records. Finally, no sample size can repair a biased sampling frame — representativeness comes from how you sample, not how many you sample.

🏁 Conclusion

Sample size determination comes down to one honest question: how wrong can you afford to be? Translate that tolerance into a margin of error (or a minimum detectable effect), fix your confidence level or power, and the formulas do the rest. For surveys, the Cochran formula with p = 0.5 gives the safe answer — 385 completes for ±5% at 95% confidence — and the finite population correction earns real savings only for small populations. For measurements, sample size scales with (zσ/e)². For group comparisons, power analysis rules, and the effect size you choose is the assumption that deserves the most scrutiny.

Keep the three levers straight: margin of error is quadratic and dominates the cost; confidence level is a multiplier you rarely change; population size hardly matters once it is large. Then remember what the number cannot do — it cannot fix a biased frame, a leading questionnaire, or a lazy sampling method. A smaller random sample beats a larger convenience sample every time.

This calculator gives you the complete workflow: survey, mean, and power modes, Slovin's formula for comparison, non-response inflation, a lookup table of the classic scenarios, four trade-off charts, and methods-section sentences ready to paste. Decide your precision, press Calculate, and recruit with confidence — that is all sample size planning ever asks of you.

❓ Frequently Asked Questions

📚 References

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