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.
**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.
**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.
**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.
**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).
Calculate the margin of error for surveys and polls.
Calculate the confidence interval for a population mean.
Calculate the Pearson correlation coefficient (r) from paired data.
Perform a chi-square goodness-of-fit test comparing observed vs expected frequencies.
Perform a one-sample t-test comparing a sample mean to a population mean.
Compare means of 3 groups using one-way analysis of variance (ANOVA).
Use 50% if unknown
Required Sample Size
385
Infinite Pop. Sample Size
385
Before finite correction
Z-Value
1.960
Effective Margin
± 4.99%