Sample Size Calculator

What This Calculator Does

This sample size calculator determines how many observations, survey responses, or experimental subjects you need for your results to be statistically meaningful. It handles two core use cases: proportion-based studies where you measure percentages, and mean-based studies where you measure averages. Both modes support finite population correction — so if your total population is 500 employees rather than 500,000 city residents, the required sample shrinks proportionally.

Enter your population size, confidence level, and margin of error, and get a statistically valid sample size instantly. No statistics degree required, no guessing, just a clear number you can use right away.

Surprising fact: Whether you are surveying 1,000 people in your town or 300 million people in the entire United States, you only need about 384 responses to get a result that is accurate to within plus or minus 5 percentage points, 95% of the time. The population size barely matters once it gets above a few thousand.

How to Use the Calculator

Step 1 — Enter your population size. This is the total number of people (or units) in the group you are studying. If your population is very large or unknown, leave it blank — the calculator assumes an effectively infinite population and returns a conservative estimate.

Step 2 — Set your confidence level. The standard choice is 95%, meaning that if you repeated this study 100 times with different random samples, approximately 95 of those samples would produce results within the margin of error you specified. Common alternatives are 90% (faster, less precise) and 99% (slower, more precise).

Step 3 — Set your margin of error. This is the maximum acceptable difference between your sample result and the true population value. A 5% margin of error on a survey showing 60% approval means the real number is between 55% and 65%. Tighter margins require larger samples.

Step 4 — Enter your expected proportion (optional). If you have a prior estimate of what the result will be, enter it. If you have no prior data, use 0.50, which produces the largest (most conservative) sample size.

Step 5 — Read your result. The calculator returns the minimum number of completed responses you need. If you expect some non-responses, divide this number by your anticipated response rate to get the number of invitations to send.

The Sample Size Formula Explained

The standard formula for determining sample size for a proportion is: n = (Z² × p × (1 − p)) / E²

Where n = required sample size, Z = z-score corresponding to your confidence level (1.645 for 90%, 1.96 for 95%, 2.576 for 99%), p = estimated proportion (use 0.5 if unknown), and E = margin of error expressed as a decimal (e.g., 0.05 for ±5%).

When the population is finite (and relatively small), a correction factor reduces the sample size: n_adjusted = n / (1 + ((n − 1) / N)) where N is the total population size. This correction becomes negligible when N exceeds roughly 20,000 — beyond that point, population size barely affects the required sample.

Worked Example: Survey of 10,000 Customers

You manage a customer database of 10,000 people and want to survey them about satisfaction with a new product feature. You want 95% confidence with ±4% margin of error. You have no prior data, so you use p = 0.50.

Uncorrected sample size: n = (1.96² × 0.50 × 0.50) / 0.04² = 0.9604 / 0.0016 = 600.25 → round up to 601.

Finite population correction: n_adjusted = 601 / (1 + ((601 − 1) / 10,000)) = 601 / 1.06 ≈ 567.

Result: You need 567 completed responses. If your expected response rate is 30%, send approximately 1,890 survey invitations.

Worked Example: Small Team Assessment

You run a 120-person company and want to measure employee engagement. You need 90% confidence with ±5% margin of error.

n = (1.645² × 0.50 × 0.50) / 0.05² = 270.6 → round up to 271. n_adjusted = 271 / (1 + (270 / 120)) = 271 / 3.25 ≈ 83.

Result: You need 83 completed responses from your 120-person team. The finite population correction cut the requirement by more than two-thirds — this is where it matters most.

When Does Population Size Actually Matter?

A common misconception is that sample size must be proportional to population size. In practice, the finite population correction only makes a meaningful difference when the sample is a large fraction of the population (roughly more than 5%).

For 95% confidence with ±5% margin of error (p = 0.50): a population of 100 requires 80 responses; 500 requires 217; 1,000 requires 278; 10,000 requires 370; 100,000 requires 383; 1,000,000 requires 384; 10,000,000 requires 384. Notice how the number plateaus. Once the population exceeds a few thousand, adding more people to the population barely changes the sample size. The real drivers are confidence level and margin of error — not population magnitude.

How Confidence Level and Margin of Error Trade Off

Tightening either parameter inflates sample size — sometimes dramatically. At 90% confidence with 10% margin of error, you need only 68 respondents. At 95% confidence with 5%, you need 385. At 99% confidence with 1%, you need 16,590. Moving from 5% to 1% margin of error at 95% confidence requires 25× more respondents. Before demanding razor-thin precision, ask whether your decision actually changes at ±5% versus ±1%.

Sample Size for A/B Testing — A Different Formula

The calculator above is designed for descriptive surveys. A/B testing — where you compare two variants and want to detect a statistically significant difference — uses a different formula entirely. For A/B tests, the key inputs are baseline conversion rate, minimum detectable effect (MDE), statistical power (typically 80%), and significance level (typically 5%). A/B test sample sizes scale inversely with the square of the MDE — detecting a 1% relative change requires 100× more traffic than detecting a 10% change.

Common Mistakes to Avoid

Ignoring non-response bias. The calculator tells you how many completed responses you need. If only 25% of people you contact actually respond, your required outreach is 4× the sample size. Worse, non-respondents are often systematically different from respondents.

Using 0.50 when you have better data. The proportion p = 0.50 gives the maximum sample size because variance peaks at that value. If a previous study showed 15% or 85%, using 0.50 overestimates your need by 40–70%. Use real data when you have it.

Confusing confidence level with accuracy. A 95% confidence level does not mean your result is 95% accurate. It means that 95% of similarly constructed intervals would contain the true value.

Treating the formula as universal. The standard formula assumes simple random sampling. Stratified sampling, cluster sampling, and multi-stage designs all require adjustments — typically a design effect multiplier.

Rounding down. Always round up. A calculation yielding 384.2 means you need 385, not 384.

Sample Size for Common Research Scenarios

National opinion polls typically need 1,068 respondents (95% / ±3%). Customer satisfaction surveys need 357–381 (95% / ±5%). Employee engagement surveys need 73–357 (90% / ±5%). Academic research studies need approximately 385 for infinite populations (95% / ±5%). Website A/B tests per variant need 3,000–30,000+ (80% power / 5% MDE). Clinical trials per arm need 50–500+. Quality control inspections need 2,401 (95% / ±2%).

How Response Rate Affects Planning

Your calculated sample size is useless without a realistic response rate assumption. In-person intercept surveys get 70–80% response rates. Phone surveys (live caller) get 15–25%. Email to existing customers gets 10–30%. Email to cold lists gets 2–5%. Social media ads get 0.5–2%. Website pop-ups get 3–8%. Plan for attrition too — incomplete surveys where the respondent drops off partway further reduce your usable sample.

The Relationship Between Sample Size and Statistical Power

Statistical power is the probability that your study will detect a real effect when one exists. A study with 80% power has a 20% chance of missing a genuine difference — what statisticians call a Type II error. Power depends on four factors: sample size (larger samples detect smaller effects), effect size (larger real differences are easier to detect), significance level (stricter thresholds require more data), and variability (higher variance requires more observations). For survey research, the sample size formula implicitly builds in power through the confidence level and margin of error. For experimental designs, power analysis is a separate, mandatory step.

When You Need a Statistician Instead of a Calculator

This calculator handles straightforward scenarios — single-proportion surveys, infinite or finite populations, standard confidence levels. You should consult a statistician when your sampling design is complex (stratified, clustered, or multi-stage), you are comparing subgroups and need adequate power within each subgroup, your study involves repeated measures or longitudinal tracking, you are designing a clinical trial with regulatory requirements, your outcome variable is continuous with unknown or highly skewed distribution, or you need to account for covariates in your analysis. A calculator gives you a starting point. A statistician ensures your study design matches your research question.