Sample Size Calculator

Understanding sample size for students and professionals

Sample size determines the precision and reliability of your estimates. Too small and results are noisy; too large and resources are wasted. With a clear confidence level and margin of error, you can plan a study that’s efficient, defensible, and aligned with your goals.

Key inputs that drive sample size

• Confidence level: The probability your interval contains the true value (e.g., 95%). Higher confidence increases required sample size.
• Margin of error (e): Desired precision around your estimate. Smaller margins require larger samples.
• Expected proportion (p) or standard deviation (σ): The variability you anticipate. If p is unknown, use 0.5 for a conservative (maximum) sample size.
• Finite population correction (FPC): When the population is not huge, FPC reduces the needed sample size appropriately.
• Design effect (DEFF): Clustered or complex sampling inflates variance. Multiply by DEFF to adjust.
• Response rate: The fraction of invited participants who complete the study. Divide by this rate to estimate invitations needed.

Core formulas

• Proportion (infinite population): n₀ = (Z² · p · (1 − p)) / e².
• Mean (infinite population): n₀ = (Z² · σ²) / e².
• Finite population correction: n = n₀ / [1 + (n₀ − 1)/N].
• Design effect adjustment: n_adj = n · DEFF.
• Invitations needed: invites = n_adj / response_rate.

Best practices

• Justify assumptions: Document how you chose p or σ (historical data, pilot study, expert judgement).
• Be consistent: Keep your confidence level and margin of error aligned with organizational or regulatory standards.
• Plan for nonresponse: Use realistic response rates to avoid underpowered studies.
• Account for design: If using clusters/strata, incorporate DEFF or run a design-based calculation.

Interpreting results

The calculated sample size reflects the minimum number of completes required to meet your precision target. If your population is small, FPC reduces the requirement. If your design is complex or response rates are lower than expected, adjust upward using DEFF and response rate to maintain precision.

Frequently asked questions

HOW TO choose confidence level and margin of error?

Steps: Use 95% confidence and ±5% margin for general surveys. For regulatory or critical decisions, consider 99% or smaller margins with larger sample sizes.

HOW TO pick p (expected proportion) when it’s unknown?

Use p = 0.5 to be conservative. If prior data suggests a different rate (e.g., 0.3), enter that to potentially reduce sample size.

HOW TO apply finite population correction (FPC)?

Enter population size N. The calculator applies n = n₀ / [1 + (n₀ − 1)/N], reducing sample size for smaller populations.

HOW TO account for clustered sampling?

Set Design effect (DEFF) > 1 (e.g., 1.2–2.0) to inflate the sample size appropriately for complex designs.

HOW TO plan invitations based on response rate?

Enter an expected response rate (e.g., 0.7). The tool calculates invites = adjusted sample size / response rate.

HOW TO compute sample size for mean outcomes?

Select Mean, enter σ and margin of error in the same units as your outcome, choose confidence level, and calculate.

HOW TO format results for reports?

Use Output format = Plain, Compact, or CSV and click Copy result to paste into documents or spreadsheets.

HOW TO embed this calculator on my website?

Copy this single‑file HTML into your project, link it from navigation, or embed via iframe with a descriptive title.