Sample size calculators

Sample size calculators by goal

Pillar page of the sample size cluster: it organizes the calculators by intent (confidence-interval estimation vs. power for a hypothesis test) to avoid overlap between the individual sub-pages.

Start by deciding whether your study needs to estimate a parameter within a margin of error, or set up a hypothesis test with a target power. These are different problems that rely on different assumptions.

To estimate a parameter (confidence interval)

Use these tools when the main output will be a confidence interval for a mean, a proportion or a difference. You need to set the confidence level, the margin of error and the expected variability.

To design a hypothesis test

Use these tools when you want to detect a difference, an association or a minimum relevant effect. You need to set the significance level, the power and the effect size.

What is sample size and why does it matter?

Sample size is the number of observations you need to collect to answer a research question with a predefined level of precision and reliability. Working it out before starting the study is essential: a sample that is too small produces unreliable results (it fails to detect real effects and yields wide confidence intervals); an oversized sample wastes resources without any meaningful gain.

The exact formula varies with the goal, but in every case n grows when you demand more precision or more power, and shrinks when the expected effect is large or the variability is low. The underlying logic is always the same: keep the estimation or decision error under control at an acceptable cost.

Two main approaches

Key concepts

• Significance level (\(\alpha\)): the maximum probability of a Type I error (rejecting a true null hypothesis). The usual value is 0.05, which corresponds to a 95% confidence level.
• Power (\(1-\beta\)): the probability of detecting a real effect when it exists. The most common targets are 80% and 90%. Increasing power requires a larger sample.
• Type II error (\(\beta\)): the probability of failing to detect a real difference (false negative). It is complementary to power: \(\beta = 1 - (1-\beta)\).
• Effect size: the minimum relevant magnitude you want to detect (difference of means, difference of proportions, correlation…). The smaller the effect, the larger the sample must be.
• Margin of error (\(E\)): in estimation studies, the maximum tolerable deviation between the sample estimate and the population value.
• Variability: the expected spread of the variable (standard deviation \(\sigma\) for means, \(p(1-p)\) for proportions). More uncertain values force you to use conservative assumptions.
• Finite population: when the sample represents an appreciable fraction of the total (\(n/N > 5\%\)), a correction factor applies that reduces the required sample size.

Quick guide to planning a sample

Before computing \(n\), define precisely what you want to measure and with what guarantees. Changing the goal midway through the study invalidates the original calculation.

1. Define the primary metric before collecting data. A study with multiple primary variables requires computing \(n\) for each one and taking the largest sample, or adjusting the significance level.
2. Use conservative assumptions if you don't have pilot data. For proportions with no prior information, use \(p = 0.5\) (maximum variability). For means, look for estimates of \(\sigma\) in similar literature.
3. Account for expected dropout. Once you have \(n\), divide it by \((1 - \text{dropout rate})\). For example, if you expect 15% dropout, recruit \(n / 0.85\) subjects.
4. Always round up. Sample size is a whole number; never truncate it.
5. Document your assumptions. Record the margin of error and confidence level (for estimation), or \(\alpha\), power, expected effect and variability (for a hypothesis test), so you can justify the design and recompute it if conditions change.