Sample size calculator

What is this calculator?

A sample size calculation turns a study goal into a minimum number of analyzable observations. The goal can be precision, such as estimating a survey proportion within a chosen margin of error, or power, such as detecting a difference between two groups if that difference is truly present. Statoma separates those two planning questions because they answer different practical problems. A margin-of-error plan asks how narrow an estimate should be. A power plan asks how likely a planned hypothesis test is to detect an effect of a specified size.

The calculator uses standard introductory planning formulas for one proportion, one mean, two proportions, and two means. These formulas are intentionally transparent: they rely on normal critical values, equal allocation for two-group comparisons, and summary assumptions that can be reviewed before data collection begins. That transparency is useful in teaching, grant planning, survey design, and early study scoping, but it also means the result should be checked against the actual sampling design.

A required sample size is not the same as the number of people to invite, records to request, or responses to start. It is the number needed after missing data, exclusions, nonresponse, and eligibility screening have been handled. If a survey expects incomplete responses or a study expects dropouts, the recruitment target must be larger than the analyzable sample returned here. The more fragile the data collection process, the more important that allowance becomes.

When to use it

• Use survey proportion mode when the target is a population share, rate, or percentage and the planning question is a margin of error.
• Use survey mean mode when the target is an average and you have a planning estimate of the outcome standard deviation.
• Use two-proportion power mode when two independent groups will be compared on a binary outcome such as a response, success, or conversion rate.
• Use two-mean power mode when two independent groups will be compared on a numeric outcome and a common planning standard deviation is reasonable.
• Use a separate specialist design calculation when observations are clustered, paired, repeated over time, heavily weighted, or assigned in unequal group sizes.

The best mode follows the design, not the result you hope to see. If the final report will estimate a single population value, use a precision mode. If the final report will compare two groups with a hypothesis test, use a power mode. Mixing those goals can produce a number that looks precise but does not protect the study from being underpowered for its actual question.

How it works

For a single proportion, the calculator starts from the familiar normal-approximation margin of error formula. The planned proportion controls the variance term, the confidence level controls the critical value, and the target margin of error controls how much uncertainty is acceptable. Smaller margins and higher confidence levels require larger samples.

In this formula, p is the planning proportion, e is the target margin of error on the proportion scale, and z is the standard normal critical value for the chosen confidence level. For a mean, the same structure uses the planning standard deviation instead of the proportion variance. The margin of error must be in the same unit as the outcome.

Power calculations add a second critical value. Alpha controls how often the test is allowed to signal an effect when the null model is true. Power controls how often the test should detect the specified effect when that effect is real. For equal-size two-sample mean comparisons, the planning approximation is based on the combined alpha and power critical values, the standard deviation, and the minimum detectable difference.

Two-proportion power calculations use the same idea but replace the standard deviation with binomial variance terms for the baseline rate and comparison rate. Statoma treats the minimum detectable effect as an absolute change. For example, moving from 0.40 to 0.45 is an absolute change of 0.05. A relative lift would need to be converted to the resulting absolute change before entering it.

Worked example

Suppose a researcher wants to estimate a population proportion with a 95% confidence level and a margin of error of 0.05. If there is no reliable prior estimate, using 0.50 for the planning proportion is a conservative choice because p(1-p) is largest at 0.50. The calculator uses the 95% normal critical value and returns 385 analyzable responses for the simple random-sample approximation.

That does not mean 385 invitations are enough. If the survey team expects only half of invited people to complete the survey, the invitation count would need to be much larger. If the survey uses a complex design, the design effect may also inflate the required number. The calculator result is the clean statistical core; the operations plan still has to account for how data are actually collected.

For a power example, suppose an experiment compares two independent groups and wants to detect an absolute rate change from 0.40 to 0.45 with 80% power at alpha 0.05. The resulting per-group sample is large because a five-point absolute difference is modest relative to the variability of a binary outcome. If that sample is unrealistic, the honest next step is to revisit the effect size, allocation, outcome definition, or study design rather than reporting a small study as if it had the desired power.

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