How do I calculate the required sample size for a survey?

For a proportion survey: n = (Z²× p×(1−p)) / e², where Z = z-score for confidence level (1.96 for 95%, 2.576 for 99%), p = expected proportion (use 0.5 if unknown — maximizes required sample size), e = margin of error. For 95% confidence and ±5% margin: n = (1.96² × 0.5 × 0.5) / 0.05² = (3.84 × 0.25) / 0.0025 = 0.96 / 0.0025 = 384 respondents.

What is a confidence interval?

A confidence interval (CI) is a range of values that likely contains the true population parameter with a specified probability. A 95% CI means: if you repeated the study 100 times, approximately 95 of the resulting intervals would contain the true value. It does NOT mean there is a 95% probability that the true value lies in this specific interval. Wider CIs indicate more uncertainty (smaller samples or higher confidence levels).

What sample size do I need for a 95% confidence level with 5% margin of error?

For a simple proportion estimate with maximum variance (p=0.5): n = (1.96² × 0.5 × 0.5) / 0.05² ≈ 384. This is the commonly cited '385 rule' for national surveys. For a ±3% margin: n ≈ 1,068. For ±1%: n ≈ 9,604. The required sample size increases dramatically as margin of error decreases.

Sample Size Calculator

Find the minimum sample size for your survey or study.

Sample Size — Formula

For proportions (infinite population)

n = (z* ² × p × (1−p)) / e²
Where e = margin of error (as decimal), p = expected proportion (default 0.5).

Finite population correction

n₁ = n₀ / (1 + (n₀−1) / N) — reduces sample size when population is small.

Related Calculators

How to Determine the Right Sample Size

Sample size depends on four factors: confidence level (how sure you want to be), margin of error (acceptable precision), population proportion (estimated prevalence), and population size (finite population correction). The standard formula for proportions is n = Z² × p(1−p) / E², where Z is the z-score for your confidence level, p is the expected proportion (use 0.5 for maximum sample size when unknown), and E is the margin of error. For a 95% CI with ±5% margin: n = 1.96² × 0.5 × 0.5 / 0.05² = 384.16 → 385.

Larger samples cost more but reduce margin of error. Doubling accuracy (halving margin of error) quadruples the required sample size. Common benchmarks: surveys needing ±3% use ~1,067 respondents; ±5% needs ~385; ±10% needs ~97. For small populations, use the finite population correction: n_adjusted = n / (1 + (n−1)/N). A population of 500 needing 385 in the infinite formula only requires 218 after correction.

Sample Size for 95% Confidence (p=0.5)

Margin of Error	Required Sample	Notes
± 1%	9604	Very precise, very costly
± 2%	2401	Academic research
± 3%	1068	Quality polling standard
± 5%	385	Most surveys and studies
± 10%	97	Rough estimates only

How Sample Size Affects Study Accuracy

Sample size is one of the most consequential decisions in study design. Too small a sample produces wide confidence intervals and low statistical power, meaning real effects in the population may go undetected (a type II error). Too large a sample wastes resources and time, and can detect trivially small effects that have no practical importance. The required sample size depends on three factors: the desired confidence level (typically 95%), the acceptable margin of error (how wide the confidence interval can be), and the estimated population proportion or variance. For proportion surveys, the maximum uncertainty occurs when the population proportion is 50%, which yields the largest required sample. Using the formula n equals z-squared times p times (1 minus p) divided by e-squared, a 95% confidence interval with a 5% margin of error and unknown proportion requires approximately 385 respondents, regardless of total population size (for large populations). Power analysis for hypothesis testing also accounts for the expected effect size and acceptable Type I error rate (alpha). Researchers in medicine, social science, and quality assurance consult statisticians or use sample size calculators to plan studies that are neither underpowered nor wastefully oversized.

Sample Size Reference Table (95% Confidence, Unknown Proportion)

Margin of error	Sample size needed
±1%	9,604
±2%	2,401
±3%	1,068
±5%	385
±10%	97