Power analysis calculator
Unlimited free power calculations for means and proportions. Get achieved power and recruitment totals using a labelled normal approximation.
Power analysis
Formula and supported method
Φ(shift−zcrit)+Φ(−shift−zcrit) two-sided, Φ(shift−zcrit) one-sided. Independent effective n=1/(1/n1+1/n2); paired/one-sample effective n=n.
Normal approximation throughout; one-sided effect assumed in the prespecified alternative direction. Enter analyzable counts; attrition only adds a recruitment target. No t-based small-sample claim.
Worked example: d=.5 with 63+63 observations at two-sided alpha=.05 gives about 80% power.
Method reference · Implementation 2026-09-14
Unlimited local calculations, no login. Inputs and results are not saved or transferred to AnswerThis. Copy or download before leaving.
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How to use power analysis calculator
- Choose the study design and an assumed effect supported by prior evidence.
- Enter analyzable sample sizes, alpha, sidedness and allocation.
- Check achieved power; recruitment allowances are a separate total and do not change analyzable n.
- Copy or download the text or CSV result with its assumptions before leaving; the page does not retain it.
Power is conditional on a specified alternative
Power is the probability of rejecting the null hypothesis under a particular assumed effect, design and test rule. It is not the probability that your hypothesis is true or that a planned study will produce an important finding. Start by defining the endpoint, reference comparison, scientifically relevant difference and analysis population. Enter the number of observations expected to be available for analysis rather than the number invited or initially recruited. For independent designs, the allocation ratio determines group 2 from group 1; for paired designs, the count is the number of complete pairs.
The supported designs are two independent means, paired means, a one-sample mean, two independent proportions and a one-sample proportion. For continuous outcomes, the input effect is the absolute difference divided by a planning SD. Use a common SD for independent means and the SD of within-pair differences for paired means. For binary outcomes, provide the two probabilities directly. In one-sample designs the second mean or probability is the prespecified null reference. Assumptions should come from relevant evidence and scientific judgement; substituting an unstable observed effect after the study can make so-called observed power little more than a restatement of the p-value.
The implemented probability model
Every power result on this page is a normal approximation. For two independent groups, the noncentral normal shift is d/sqrt(1/n1+1/n2). For paired or one-sample means it is d sqrt(n). A two-sided test uses critical value z at 1−alpha/2 and adds the upper and lower rejection probabilities under the shifted distribution: Φ(shift−z)+Φ(−shift−z). A one-sided test uses z at 1−alpha and Φ(shift−z), assuming the effect lies in the stated alternative direction. The result therefore depends on sidedness even when the effect and sample sizes are unchanged.
For proportions, the standardized shift uses the magnitude of Cohen h, defined as 2 asin(sqrt(p1))−2 asin(sqrt(p2)). This variance-stabilizing scale supplies an approximate normal design; it is not an exact binomial calculation or a guarantee of accurate tail behaviour for sparse events. Means power also does not use a noncentral t distribution. If sample sizes are small or uncertainty in the estimated SD is central, compare with a dedicated t-based implementation. Significance alpha must be between zero and one. Zero effects and invalid probabilities are rejected so a nominal alpha value is not accidentally presented as useful planning power.
Explore allocation, attrition and meaningful sensitivity
Unequal allocation changes the effective information contributed by a fixed total. With a ratio of two, the tool sets group 2 to ceil(2×group 1); more generally it rounds the ratio-derived count upward to a whole observation. The power result uses those actual analyzable counts. With d=0.5, two-sided alpha=0.05 and 63 participants in each independent group, power is slightly above 0.80 under this normal approximation. Smaller effects or fewer analyzable observations lower it. Use several defensible scenarios to communicate the range of plausible operating characteristics instead of presenting a single assumed effect as certain.
The attrition field produces recruitment targets from the entered analyzable counts. Each group is inflated by dividing by one minus the attrition fraction and rounding upward. Attrition does not change the displayed achieved power because the power calculation already assumes the entered complete observations are available. If you want power under fewer completed observations, reduce the analyzable count explicitly. Loss mechanisms can introduce bias that adding participants cannot remove. The calculator does not implement cluster, survival, regression, noninferiority, equivalence, sequential or complex repeated-measure designs. It also does not correct for testing several primary outcomes. Download the result with its assumptions and discuss the relevant design with a statistician before setting recruitment or interpreting a completed study. The sample-size tab solves the inverse planning question using the same implementation.
Frequently asked questions
Does attrition reduce the power shown?
No. You enter analyzable observations. Attrition adds a separate recruitment target. To explore fewer completed observations, reduce the analyzable count.
Can I calculate observed power from my final effect?
You can enter an effect, but observed-effect power is generally uninformative after a study. Prefer an effect estimate and confidence interval for interpretation.
Does this include cluster or survival designs?
No. The forms implement only the listed simple means and proportions designs using a normal approximation. Use design-specific software for unsupported analyses.
Are calculations free and unlimited?
Yes. Local calculations have no quota and require no login. External lookups are rate limited to protect provider access; waiting and retrying never requires signup.
Does AnswerThis save or receive my result?
No. Signup opens the AnswerThis app; it does not save or transfer this result. Copy or download your calculation before leaving the page.
How can I reproduce a result?
Keep the supplied inputs, selected method, units, assumptions and implementation date in the text or CSV export. Numerical fixtures are checked against SciPy 1.15.3, statsmodels 0.14.4 and pingouin 0.5.5.
What if a value is missing or invalid?
Required fields show a specific error. Zero-variance and invalid-degree-of-freedom cases are refused. Missing provider metadata is labelled unknown or not reported, and a failed lookup is never replaced by invented data.
Sources and related research tools
Method version: 14 September 2026. Numeric reference validation: SciPy 1.15.3, statsmodels 0.14.4, pingouin 0.5.5. Numeric inputs stay in your browser; these calculations use no AI model or external calculation service.
SciPy statistical distributions; statsmodels statistical methods; OpenAlex access and limits; OpenAlex metric definitions.