There is no universal sample size that guarantees a reliable result. Choose a sample only after defining the decision it will support, the false-positive risk you can tolerate, the smallest effect worth detecting (or the precision you need), and the study design. Then calculate for those assumptions and the analysis you actually plan to use.
Start with the decision—not a target number
Ask, “How many measurements should be included in the sample?” only after specifying what those measurements must help you decide. Are you testing whether two groups differ, estimating a quantity within a given margin of error, or checking whether a system meets a fixed performance threshold? Each goal calls for different inputs and potentially a different calculation.
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Write down the population, primary outcome, parameter or estimand, comparison, and action a positive finding would trigger. A sample-size calculation for one outcome or decision rule does not automatically support other claims.
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The National Institute of Standards and Technology (NIST) puts the central limitation plainly: “Unfortunately, there is no correct answer without additional information (or assumptions).” For a mean-based problem, relevant inputs include the planned significance level, the desired power for a specified alternative, and the population standard deviation. The appropriate inputs change with the question and design. NIST: Sample sizes required
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Set the false-positive tolerance before collecting or reviewing results
In a planned statistical test, alpha is the Type I error risk: the procedure’s probability of rejecting its null hypothesis when that null is true, under the assumptions of the test and design. It is not the probability that a particular positive finding is false. That probability also depends on matters such as how common real effects are, study quality, selection, and analytical flexibility; alpha alone does not quantify it.
Choose alpha in light of the consequence of a false claim, relevant standards, and the precise family of hypotheses and decision rule. State which primary endpoint and analysis it applies to before looking at results. Raising the sample size does not, by itself, lower alpha; alpha is a planned property of the testing procedure.
Account for every route to a positive claim
If a study can declare success based on any of several endpoints, subgroups, interim looks, or analyses, the chance of at least one false-positive conclusion can increase if those routes are treated as independent opportunities without appropriate control. Decide in advance how claims will be grouped and ordered and what multiplicity strategy will govern them.
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There is no universally best correction: the right strategy depends on the study objectives and decision rule. The FDA’s October 2022 guidance discusses endpoint grouping, ordering, and recognized strategies for multiplicity in clinical trials of human drugs and biological products. It is specific to that setting, not a general regulatory rule for every kind of study. FDA: Multiple Endpoints in Clinical Trials
Define what is worth detecting—or how precise the estimate must be
For a power-based test
Specify the smallest effect that would change a real decision. This is the meaningful alternative for the calculation; choosing a smaller or larger effect just to obtain a convenient sample size can produce a study that does not answer the practical question. Then set a target power, 1−β, for detecting that effect at the chosen alpha. Power is conditional on the effect, test, design, and outcome model—not a guarantee about the result of a particular study.
If missing a meaningful effect would be especially costly, choose a lower acceptable β (and thus higher target power). All else equal, that generally requires more observations. NIST lists alpha, beta for a specified alternative, and variability among the assumptions needed for sample-size planning. NIST: Sample sizes required
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For an estimate or performance threshold
If the goal is estimation, specify the maximum uncertainty or interval width you can accept rather than choosing power for a hypothesis test that is not your objective. If the aim is to show that a binary-response system meets a fixed performance threshold, the threshold and the acceptable risk or required confidence are central inputs. NIST Technical Note 2045 addresses that threshold-testing setting; its method is not interchangeable with a general test of two means. NIST TN 2045: Confirming a Performance Threshold with a Binary Experimental Response
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There is no single formula for every study. The calculation must reflect both what is measured and how observations are collected. Means, proportions, binary performance thresholds, clustered measurements, repeated measures, and unequal group allocation can require different methods. A calculation that ignores dependence within clusters, for example, may overstate how much information the sample provides.
Use defensible estimates of variability or baseline event rate, and specify the one- or two-sided test, allocation ratio, dependence structure, missingness or attrition assumptions, and planned analysis. Prior information and stratification can also affect sample requirements; include them only when justified for the study in question. NIST’s planning guidance frames sample selection in terms of the question, variability, precision, practical constraints, and cost. NIST: Selecting Sample Sizes
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One conditional proportions example
NIST’s proportions example gives approximately 102 observations under its stated one-sided assumptions; using a continuity correction changes the example to 112. These are outputs for that particular set of assumptions and method, not default sample sizes. Check the inputs and method before applying an example to another study. NIST: Sample sizes required
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Stress-test the calculation and check feasibility
A sample-size result is only as useful as its assumptions. Where nuisance inputs such as variance, event rate, dependence, or missingness are uncertain, examine plausible scenarios rather than relying on one optimistic estimate. Confirm that the sample-size method matches the final analysis. Complex or adaptive designs may warrant simulation and specialist statistical review; the FDA’s Bayesian clinical-trial design guidance, for example, emphasizes assessing plausible scenarios and reporting operating characteristics. FDA: Guidance for Industry on the Use of Bayesian Statistics in Medical Device Clinical Trials
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Finally, weigh the burden of recruiting, measuring, and analyzing more observations against the value of the decision and the risks of both false alarms and missed effects. NIST Technical Note 2118 illustrates this trade-off in the specialized context of radiation-detection false-alarm testing; its domain-specific procedures should not be treated as a general study-size formula. NIST TN 2118: False Alarm Testing for Radiation Detection Systems
Report enough detail for the calculation to be checked
A sample-size justification should let another reader understand what question the number answers and reproduce the main assumptions. Report the primary endpoint, target effect or precision, alpha, target power where relevant, variability or baseline rate, study design and allocation, multiplicity strategy, planned analysis, and any allowance for missing or unusable observations. ARRIVE’s sample-size guidance, written for research involving animals, likewise calls for justification tied to the research question and a predefined meaningful effect when power is used. ARRIVE: Sample size
- False-positive control: Which exact hypotheses or family of claims does the decision rule cover?
- Power or precision: What minimum meaningful effect or acceptable uncertainty does the design target?
- Design fit: Do the calculation and final analysis account for the outcome type, allocation, clustering, repeated measures, and missingness?
- Robustness and burden: How much does the result change under plausible assumptions, and is the recruitment or measurement burden feasible?
A larger sample can reduce the chance of missing a specified effect under a specified design. It cannot fix biased sampling, a poorly chosen decision rule, unplanned endpoint fishing, or a mismatch between the calculation and the final analysis.
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