Configure your study assumptions
Use standardized effects or derive them from practical input values.
Example study assumptions
This example compares two independent means with a standardized effect of 0.50, 80% power, and a two-sided 5% alpha level.
| Input | Example value | Why it matters |
|---|---|---|
| Design | Two independent means | Compares outcomes from two separate groups. |
| Mean difference | 5.0 units | Defines the practically meaningful difference. |
| Standard deviation | 10.0 units | Converts the raw difference into Cohen’s d = 0.50. |
| Alpha | 0.05, two-sided | Sets the false-positive threshold. |
| Target power | 0.80 | Sets the probability of detecting the planned effect. |
| Attrition | 10% | Increases enrollment beyond analyzable observations. |
Formula used
Required sample size for one mean or paired mean:n = (Zα + Zβ)² / d²
Required Group 1 size for two independent groups:n₁ = (Zα + Zβ)² × (1 + 1/r) / d², where r = n₂ / n₁.
One-proportion and two-proportion designs:
The calculator uses Cohen’s h = 2 asin(√p₁) − 2 asin(√p₀) in the same normal-approximation framework.
Correlation designs:
The calculator uses Fisher’s separation z = atanh(ρ₁) − atanh(ρ₀) and n = 3 + (Zα + Zβ)² / z².
For two-sided tests, Zα is based on alpha ÷ 2. For one-sided tests, it is based on alpha. The approximations are most reliable for moderate sample sizes and planned effect assumptions.
How to use this calculator
- Choose the result you need: sample size, achieved power, or minimum detectable effect.
- Select a study design that matches the primary comparison.
- Set alpha, alternative direction, and desired power before reviewing outcomes.
- Enter a manual effect or use raw means, proportions, or correlations.
- For two-group studies, specify the expected allocation ratio or both existing group sizes.
- Apply expected attrition and recruitment rounding when estimating final enrollment.
- Review the table and graph, then export the calculation for your protocol or review notes.
Plan studies with statistical power
Why power matters
Statistical power measures a study’s ability to detect a real effect. A well planned study reduces wasted time, money, and participant effort. This calculator helps you connect sample size, effect size, significance level, and desired power before data collection begins.
Power is commonly set at 80% or 90%. An 80% target accepts a 20% chance of missing an effect of the chosen size. Higher power usually needs more observations. Smaller effects also need larger samples. Your assumptions should come from prior studies, pilot data, or practical importance.
Match the design to the question
Choose a test design that matches the research question. Use one mean for a single group against a benchmark. Use paired means for before-and-after measurements. Use two independent means when groups contain different people. Select proportion designs for binary outcomes. Select correlation when studying linear association between two continuous measures.
Effect sizes and planning formulas
The calculator uses normal-approximation planning formulas. For mean comparisons, Cohen’s d represents the difference relative to standard deviation. For proportions, Cohen’s h uses an arcsine transformation. For correlations, Fisher’s z transformation stabilizes the sampling distribution. These measures let designs share a clear planning framework.
Select required sample size to calculate the number of observations needed. Choose achieved power to assess a planned enrollment. Choose minimum detectable effect to find the smallest standardized difference your sample can reasonably identify. The allocation ratio lets two groups have unequal sizes. Attrition adjustment increases enrollment so completed data remain adequate.
Choose defensible assumptions
Two-sided testing is appropriate when effects in either direction matter. One-sided testing can use fewer observations, but it requires a justified direction before data are seen. Do not choose it merely to lower the required sample. Enter a realistic alpha level and preserve that choice in your analysis plan.
Results are estimates, not guarantees. Real studies can lose precision through missing data, clustering, covariate adjustments, non-normal outcomes, or changed variances. Consider inflating enrollment for expected exclusions. Consult a statistician for complex designs, repeated measures, survival outcomes, or multiple comparisons.
Use results as a planning record
Use the power curve to see how enrollment changes detection probability. Download the results for protocol notes or team review. Recalculate whenever the design, expected effect, or practical constraints change. Careful planning makes final conclusions more credible and easier to defend. It supports transparent discussions among investigators, reviewers, funders, and ethics committees when resource decisions need clear explanation.
Frequently asked questions
1. What is statistical power?
Statistical power is the probability of detecting a real effect of the chosen size when it exists. Power equals one minus the Type II error rate. Higher power reduces the chance of a false negative.
2. Is 80% power always enough?
No. Eighty percent is common, but 90% or higher may be appropriate when missed effects are costly, recruitment is feasible, or results will guide important decisions. The target should be justified before data collection.
3. What is Cohen’s d?
Cohen’s d expresses a mean difference in standard deviation units. A difference of five with a standard deviation of ten gives d = 0.50. It helps compare effects across measures with different units.
4. Why do smaller effects need more participants?
Small effects produce weaker signals relative to random variation. More observations reduce sampling error, making the signal easier to distinguish from noise. This increases the chance that the planned test reaches significance.
5. Should I use a one-sided test?
Use a one-sided test only when the opposite direction would not alter the scientific conclusion and the direction is fixed beforehand. Choosing it after seeing data is not a defensible planning decision.
6. How does attrition affect sample size?
Attrition reduces the analyzable sample. The calculator divides the required analyzable count by the expected retention rate, then rounds upward. Enter a realistic loss percentage based on similar studies or operations data.
7. What does the allocation ratio mean?
The allocation ratio is Group 2 divided by Group 1. Equal groups use 1.00. Unequal groups can be necessary for cost, availability, or ethics, but they usually increase the total sample needed.
8. Can this calculator handle paired data?
Yes. Choose paired mean difference for matched participants or repeated measurements. Enter the expected mean change and the standard deviation of the paired differences, not the separate baseline and follow-up standard deviations.
9. Are the proportion formulas exact?
They are planning approximations based on Cohen’s h and normal theory. They are useful for moderate samples and probabilities away from the extremes. Exact or continuity-corrected methods may be preferable in specialized situations.
10. What is minimum detectable effect?
It is the smallest effect size that the planned sample can detect at the specified alpha and power. It helps judge whether an existing enrollment can answer the question with useful precision.
11. When should I consult a statistician?
Consult a statistician for clustered trials, repeated measures, survival analysis, covariate adjustment, unequal variances, noninferiority, multiple endpoints, adaptive studies, or formal regulatory submissions. These designs often need methods beyond simple normal approximations.