How does population size affect sample size?

For large populations (over 100,000), population size has minimal impact. For smaller populations, a finite correction factor reduces the required sample size. For a population of 1,000 needing ±5% precision, raw n = 385 drops to ~278 after correction.

Why does a lower margin of error require a larger sample?

A smaller margin of error means more precision, which requires more data points. The relationship is quadratic: halving the margin of error quadruples the required sample size. Going from ±5% to ±2.5% takes ~385 to ~1,537 respondents.

Why use 50% as expected proportion when unknown?

The variance term p(1−p) is maximized at p = 0.5 (yielding 0.25). Using 50% gives the most conservative — largest — sample size, guaranteeing precision regardless of the true proportion. If you know p is much smaller (say 5%), the required n drops substantially.

How many respondents do national polls use?

Most reputable national polls use 800-1,500 respondents, giving margins of error around ±3% at 95% confidence. This matches our calculation: ±3% at 95% with p = 0.5 requires ~1,068 respondents.

Do I need different formulas for different research questions?

Yes. Estimating one proportion uses the formula here. Estimating a mean needs σ instead of p(1−p). Comparing two groups (A/B tests, clinical trials) needs power analysis with both significance level and power. Each scenario uses different math.

What if I have a budget constraint instead of a precision goal?

Reverse the calculation. Given your maximum n, compute the achievable margin of error: E = z × √(p(1−p)/n). For n = 200 at 95%: E = 1.96 × √(0.25/200) = ±6.9%. Decide whether that precision is sufficient.

Should I add a buffer for non-response?

Yes. Calculate the required completed responses, then inflate based on expected response rate. For an online survey with 20% response rate needing 400 completed responses, send invitations to 400 / 0.20 = 2,000 people. Add another 10-20% for ineligibility.

Does this calculator work for A/B testing?

It estimates sample size for a single proportion. A/B tests comparing two proportions need power-based formulas accounting for both groups and your minimum detectable effect. Use tools like G*Power or statsmodels for proper A/B test sizing — A/B tests typically need 4-10× more sample than single-proportion estimation.

Sample Size Calculator

Determine how many survey responses you need to achieve a desired margin of error at a given confidence level. Optionally apply a finite population correction.

Sample size determines how precisely a survey, study, or experiment can estimate a population parameter. Too few responses produce wide error margins and unreliable conclusions; too many waste budget and time. A sample size calculation answers the fundamental planning question: how many people do I need to survey, how many users do I need in an A/B test, or how many patients must I enroll in a trial to detect a meaningful effect?

This calculator returns the minimum sample size for estimating a proportion (e.g., percentage of voters favoring a candidate, defect rate in a process, conversion rate in a marketing test) at your chosen confidence level and margin of error. It optionally applies a finite population correction (FPC) when the population is small enough that sampling materially reduces the required n.

The math behind sample size is the inverse of the margin-of-error formula. Where margin of error tells you how precise your current sample is, sample size tells you how large a sample you need to reach a target precision. The relationship is roughly: halving the margin of error quadruples the sample size. Going from ±5% to ±2.5% precision typically means 4× the respondents.

Common applications: political polling, market research, customer satisfaction surveys, public-health studies, quality-control sampling, A/B testing planning, academic research, and any planning task where the cost of additional observations must be balanced against statistical precision.

Inputs

Desired Margin of Error (%)

Confidence Level

Expected Proportion (%)

Use 50% if unknown

Population Size (0 = infinite)

Results

Required Sample Size

385

Infinite Pop. Sample Size

385

Before finite correction

Z-Value

1.960

Effective Margin

± 4.99%

Last updated: May 29, 2026

Formula

**Sample size for a proportion (infinite population):** n = (z² × p × (1 − p)) / E² Where: - z = z-score for the chosen confidence level - p = expected proportion (use 0.5 if unknown — gives the maximum, most conservative n) - E = desired margin of error (as a decimal) **Z-scores for common confidence levels:** | Confidence | Z | |---|---| | 80% | 1.282 | | 90% | 1.645 | | 95% | 1.960 | | 99% | 2.576 | **Worked example: political poll** Goal: estimate vote share to ±3% at 95% confidence. Unknown proportion → use p = 0.5. n = (1.96² × 0.5 × 0.5) / 0.03² n = (3.8416 × 0.25) / 0.0009 n = 0.9604 / 0.0009 n ≈ 1,068 Round up: **1,068 respondents** needed. **Finite population correction (FPC):** For smaller populations N, the required sample shrinks: n_adjusted = n / (1 + (n − 1) / N) Example: N = 5,000, n_infinite = 1,068. n_adj = 1,068 / (1 + 1,067/5,000) n_adj = 1,068 / (1 + 0.2134) n_adj = 1,068 / 1.2134 n_adj ≈ 880 **Sample size for estimating a mean:** n = (z × σ / E)² Where σ = population standard deviation, E = margin of error in original units. Example: estimating mean income to ±$500 at 95% confidence with σ = $10,000: n = (1.96 × 10,000 / 500)² = 39.2² ≈ 1,537 **Sample size for two-proportion comparison (A/B testing):** n per group ≈ (z_{α/2} + z_β)² × (p₁(1−p₁) + p₂(1−p₂)) / (p₁ − p₂)² Where z_β corresponds to desired power (typically 80% → z_β = 0.84). Example: detect lift from 5% baseline to 6% conversion (relative 20% lift), 95% confidence, 80% power: n ≈ (1.96 + 0.84)² × (0.05 × 0.95 + 0.06 × 0.94) / 0.01² n ≈ 7.84 × (0.0475 + 0.0564) / 0.0001 n ≈ 7.84 × 0.1039 / 0.0001 n ≈ 8,146 per variant **Quick reference table (95% confidence, p = 0.5, infinite N):** | Margin of error | n | |---|---| | ±1% | 9,604 | | ±2% | 2,401 | | ±3% | 1,068 | | ±4% | 600 | | ±5% | 385 | | ±7% | 196 | | ±10% | 96 | **Rule of thumb: doubling precision quadruples cost.**

How to use this calculator

Enter desired margin of error as a percentage (e.g., 5 for ±5%).
Choose confidence level (95% is the standard).
Enter expected proportion if known; use 50% for the most conservative estimate.
Enter population size if it is small (under ~100,000); otherwise leave at 0 for infinite.
Calculator returns minimum respondents required.
Add 10-20% buffer to account for non-response and ineligible respondents.

Worked examples

National political poll

**Scenario:** Pollster wants to estimate presidential approval to ±3% at 95% confidence. National adult population ≈ 250 million (treat as infinite). Unknown proportion → p = 0.5. **Calculation:** n = (1.96² × 0.5 × 0.5) / 0.03² = 0.9604 / 0.0009 ≈ 1,068. **Result:** Need ~1,068 completed interviews. With a typical 30% response rate, plan to contact ~3,600 voters. This matches the n ≈ 1,000 you see reported in major national polls.

Small-town satisfaction survey

**Scenario:** Town of 4,000 residents wants ±5% precision at 95% confidence on a satisfaction question. **Calculation:** Unadjusted n = (1.96² × 0.25) / 0.05² ≈ 385. Apply FPC: n_adj = 385 / (1 + 384/4000) = 385 / 1.096 ≈ 351. **Result:** Survey 351 residents — about 9% of the town. The finite-population correction saved 34 surveys vs. the infinite-population estimate.

A/B test planning

**Scenario:** E-commerce site has 5% baseline conversion. Wants to detect a 1-percentage-point lift (5% → 6%) at 95% confidence and 80% power. **Calculation:** Using two-proportion formula: n per group ≈ 8,146. **Result:** Need ~8,150 visitors per variant — about 16,300 total. At 1,000 visitors/day, the test runs ~16 days. If only 4,000 visitors/day are available, plan for ~4 days. Stopping early on apparent winners inflates false-positive rates — wait for the planned sample.

When to use this calculator

**Use sample-size calculations to plan:**

- **Surveys and polls**: determine respondents needed for target precision. - **A/B tests**: determine traffic per variant before launching. - **Clinical trials**: determine patients per arm to detect a clinically meaningful effect. - **Quality control**: determine inspection sample size for a process. - **Customer research**: determine review counts needed to estimate satisfaction. - **Academic studies**: justify n in grant applications and IRB submissions.

**Key inputs to choose carefully:**

- **Margin of error**: how precise must the answer be? Tighter precision = larger n (quadratic relationship). - **Confidence level**: 95% is standard. 99% requires ~70% more sample; 90% needs ~30% less. - **Expected proportion**: if completely unknown, use 50% (maximum variance). If you have prior estimates, use them — extreme proportions need smaller n. - **Population size**: matters only when small (under ~10,000) relative to sample. Otherwise treat as infinite. - **Power (for tests)**: 80% is convention; clinical trials often require 90%.

**Real-world adjustments:**

- **Non-response**: typical online surveys 5-30% response, phone 10-20%, mail 20-40%. Inflate contact list accordingly. - **Ineligibility**: some respondents won't qualify. Add 10-25% buffer. - **Stratified sampling**: if subgroup analysis is needed, calculate n per stratum separately. - **Multiple comparisons**: testing 10 hypotheses at α = 0.05 inflates false-positive rate; use Bonferroni or FDR correction and recalculate n.

**Common contexts:**

- **Political polling**: ±3% at 95% → ~1,000. - **Market research**: ±5% at 95% → ~400. - **Customer satisfaction**: ±5% at 95% → ~400. - **Manufacturing QC**: lot-acceptance sampling tables (ANSI/ASQ Z1.4). - **Clinical trials**: power analysis with effect size, often hundreds to thousands per arm.

**Software:**

- **R**: pwr package (proportions, t-tests, ANOVA). - **G*Power**: free desktop tool, widely cited in clinical literature. - **Python**: statsmodels.stats.power. - **SAS/SPSS**: built-in power and sample size modules.

**Pitfalls:**

- **Underestimating drop-out**: trials lose 10-30% of enrollees; inflate n. - **Optimistic effect sizes**: small estimated effects need huge n. Be honest. - **Ignoring clustering**: classroom or clinic-level sampling needs design effect adjustment (often 1.5-3× n). - **Single estimate when comparing groups**: comparison tests need substantially more n than estimation. - **Stopping early without planned interim analysis**: inflates type-I error.

**Don't oversize either:** unnecessarily large samples waste resources, prolong studies, and may be unethical in clinical contexts (exposing more subjects than needed to risk).

Common mistakes to avoid

Using estimation sample size for comparison tests (A/B tests need more).
Forgetting to inflate for non-response (final n much smaller than contacted).
Using 50% proportion when known proportion is much smaller (overestimates n).
Ignoring population size for small populations (overestimates n).
Failing to plan for subgroup analyses (each subgroup needs its own n).
Treating margin of error as the same as confidence interval width (it is half-width).
Stopping data collection early when results look favorable (inflates false positives).
Confusing 95% confidence with 95% power (different concepts).

Frequently Asked Questions

Sources & further reading

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