Sample Size Calculator

Confidence Level

Margin of Error (%)

Population Proportion (%)

Use 50% if unsure (maximizes sample size)

Population Size

Enter 0 for infinite/unknown population

Required Sample Size

383responses

CL95%MoE±5%Pop100,000

Sample Size Scenarios

Compare CL × MoE trade-offs

CL \ MoE	±1%	±2%	±3%	±5%	±10%
90%	6,336	1,663	746	270	68
95%	8,763	2,345	1,056	383	96
99%	14,228	3,982	1,810	660	166

Formula Breakdown

Step-by-step calculation with your values

z = 1.9600 (for 95% confidence)

p = 0.50, e = 0.0500

n₀ = z² × p(1-p) / e² = 385

n = n₀ / (1 + (n₀-1)/N) = 383 (N = 100,000)

Using Cochran's formula with finite population correction.

How the Sample Size Calculator Works

Three modes for surveys, A/B tests, and reverse calculations

Determining the right sample size is the foundation of any statistically valid survey, research study, or experiment. Too few responses and your results lack precision; too many and you waste time and budget.

Survey Mode

Find minimum sample size for a given confidence level and margin of error

A/B Test Mode

Calculate visitors per variation using power analysis

MoE Mode

Reverse-calculate precision from an existing sample

Sample Size Formulas Explained

The math behind each calculation mode

n₀ = z² × p(1-p) / e²

Cochran's formula — z is the z-score for your confidence level, p is the expected proportion, e is the margin of error

n = n₀ / (1 + (n₀-1) / N)

Finite population correction — reduces sample when population N is small and known

n = (z_α/2 + z_β)² × (p₁(1-p₁) + p₂(1-p₂)) / (p₂-p₁)²

Two-proportion z-test — p₁ is baseline, p₂ is expected with MDE, z_β from power

Z-Score Reference

90% → z = 1.64595% → z = 1.96098% → z = 2.32699% → z = 2.576

Common Sample Sizes at a Glance

Assumes 50% proportion, large population

Quick Poll

90% CL, ±10%

385

Standard Survey

95% CL, ±5%

1,068

Market Research

95% CL, ±3%

16,590

High-Precision

99% CL, ±1%

Key insight: Halving the margin of error roughly quadruples the required sample. Going from ±5% (385) to ±2.5% needs ~1,537 responses. The cost of precision grows exponentially.

Practical Examples

Real-world sample size calculations

Customer Satisfaction Survey

10,000 users • 95% CL • ±3% MoE • 40% response rate

965 responses needed → send to 2,413 users

E-Commerce A/B Test

5% baseline • 20% relative MDE • 95% sig • 80% power

~8,200 per variation → 16 days at 1K daily visitors

Employee Engagement Survey

350 employees • 95% CL • ±5% MoE

184 responses (FPC reduces from 385 to 184)

Clinical Research Trial

99% CL • ±2% MoE • 15% dropout buffer

4,148 participants → recruit ~4,975

Who Uses a Sample Size Calculator?

From market research to clinical trials

Market Researchers

Justify methodology in proposals; funding agencies expect formal sample size justification

Product & Growth Teams

Determine A/B test duration; avoid under-powered tests that miss real improvements

QA & Audit Professionals

Statistical sampling for inspecting transactions per ISO 2859 and ANSI/ASQ Z1.4

Academic Researchers

Thesis sample sizing, experiment design, and IRB submissions with power analysis

Common Mistakes When Calculating Sample Size

Pitfalls that lead to unreliable results

Not accounting for non-response

Divide required sample by expected response rate. Need 400 at 25% rate → invite 1,600.

Stopping A/B tests early

Peeking at results inflates false positives. Pre-commit to the calculated sample size.

Confusing confidence level & interval

Level (95%) = method reliability. Interval (±5%) = result precision. Margin of error = half the interval width.

Always using 50% proportion

If prior data shows ~25%, use it. Reduces sample by 25-40% vs. the worst-case 50% estimate.

Unrealistic minimum detectable effect

Detecting a 5% lift needs ~16× more traffic than a 20% lift. Be realistic about achievable improvements.

Frequently Asked Questions

Common questions and detailed answers

For most surveys, 95% confidence with ±5% margin of error is the standard starting point, which requires about 385 responses for large populations. For market research or important business decisions, ±3% margin of error (about 1,068 responses) is recommended. Quick internal polls can use 90% confidence with ±10% (about 68 responses). The right sample size depends on how precise you need your results to be and what decisions rest on the data.

To calculate sample size for research: (1) Choose your confidence level — 95% is standard for most fields, 99% for medical research. (2) Set your margin of error — ±5% for exploratory studies, ±2-3% for definitive studies. (3) Estimate the population proportion from prior research or use 50% if unknown. (4) Enter your population size if finite. The calculator applies Cochran's formula: n₀ = z² × p(1-p) / e², with finite population correction when applicable.

For populations above about 50,000, population size has almost no effect on the required sample. The finite population correction (FPC) only makes a meaningful difference for small populations. For example, at 95% confidence with ±5% margin of error: a population of 100 needs 80 responses, 500 needs 217, 1,000 needs 278, 10,000 needs 370, and 100,000+ needs 385. The correction formula is n = n₀ / (1 + (n₀-1)/N).

The confidence level (e.g., 95%) indicates how often the method produces an interval that contains the true population value — if you repeated the study 100 times, about 95 of those intervals would contain the true value. The margin of error (e.g., ±5%) is the range around your sample result. Together they define precision: '95% confident the true value is within ±5% of our result.' Higher confidence levels and smaller margins both require larger samples.

The required visitors depend on your baseline conversion rate, the minimum detectable effect (MDE), significance level, and statistical power. For a typical website with 5% conversion rate wanting to detect a 20% relative improvement (to 6%): about 8,200 visitors per variation at 95% significance and 80% power. Lower baselines need much more traffic — a 3% baseline with the same 20% MDE needs about 14,000 per variation. Always calculate before starting — stopping tests early inflates false positives.

Statistical power (typically set at 80%) is the probability that your test will correctly detect a real effect when one exists. It equals 1 minus the Type II error rate (beta). At 80% power, there's a 20% chance you'll miss a real improvement and incorrectly conclude 'no effect.' Increasing power to 90% reduces this risk but requires about 30% more traffic. Industry standard is 80% — the trade-off between catching real effects and practical sample size requirements.

The MDE is the smallest relative change your test can reliably detect. To choose it: (1) Calculate the business impact — if a 10% lift in conversion means $100K/year, that's worth detecting. (2) Consider practical significance — even a statistically significant 1% lift may not justify implementation costs. (3) Be realistic — most A/B tests produce 5-15% lifts. A common starting point is 20% relative MDE, which balances detection sensitivity with feasible sample sizes.

The formula's variance term p×(1-p) reaches its maximum value of 0.25 at p = 50%, making the calculated sample size the largest possible. Any other proportion (30%, 70%, 10%) produces a smaller variance and smaller required sample. By using 50%, you guarantee your sample is large enough regardless of the actual proportion. If you have prior data (e.g., from a pilot study showing 25% approval), use that value for a more efficient, smaller sample.

Divide the calculated sample size by your expected response rate to get the number of invitations needed. For example: 400 required responses ÷ 0.30 response rate = 1,334 invitations. Typical response rates: email surveys 20-30%, phone surveys 10-20%, in-person 60-80%, employee surveys 50-70%. Always overestimate slightly — it's better to have more responses than fewer. You can also add 10-15% buffer for incomplete or unusable responses.

This calculator handles proportions (e.g., percentage who approve, conversion rate). For means (e.g., average spending, test scores), the formula uses standard deviation instead of p(1-p): n = (z × σ / e)². If you need to estimate a mean, you'll need an estimate of the population standard deviation from prior studies or a pilot test. The overall logic is the same — larger samples give more precise estimates.

This calculator works for basic clinical sample size estimation using proportions (e.g., treatment response rates). For clinical trials, researchers typically use 99% confidence with ±1-2% margin of error, and add 15-20% for patient dropout. However, complex trial designs (crossover, factorial, adaptive) may require specialized software like G*Power or nQuery. Always consult a biostatistician for regulatory submissions — agencies like the FDA require formal power analysis documentation.

Sample size directly affects your ability to achieve statistical significance. Larger samples produce narrower confidence intervals and smaller p-values, making it easier to detect real effects. However, with very large samples, even trivially small differences become 'statistically significant.' This is why it's important to also consider practical significance — the effect size that actually matters for your decision. The ideal approach is to pre-determine your minimum detectable effect and calculate the sample size needed to detect it.

Embed Sample Size Calculator

Add this calculator to your website or blog for free.

Related calculators from other categories

Last updated Apr 4, 2026