Power is the chance of detecting a real effect — and it depends on how large the effect really is.
What is this about?
A study with "80% power" detects a real effect in 8 of 10 cases — but only for one specific effect size. If the true effect is smaller, the chance drops; if it is zero, it only equals the false-alarm level α. Power depends on effect size, sample size, significance level and test direction. That is why it is a curve, not a single number. The example in the tool: a clinical study on an antidepressant plans with d = 0.50 from the literature and n = 64 per group (80% power) — but a pilot study only points to d ≈ 0.30.
How to proceed
First the sliders on the left: significance level α, test direction (two- or one-sided), design (1 or 2 groups), n, planned effect size d and target power. 1. Power is a curve — not a value — see how the power curve shifts with n, and how the H₀ and H₁ distributions overlap. 2. The power landscape — n × d × power as a heatmap: where is the region with sufficient power? 3. What if the true δ is smaller? — sensitivity analysis: the power at your n for every possible true d.
What to watch for
Power holds only for the assumed effect size. If it is estimated too optimistically from the literature or a pilot study, the actual power is often much lower than planned. Small effects need disproportionately many people: an effect half as large requires roughly four times as many for the same power. With a true effect of zero only the false-alarm rate α remains as "power".
Things to try
Set d from 0.50 to 0.30 and watch the power at n = 64 collapse. Then increase n until the target power is reached again and check in the heatmap how large the effort becomes.
What's next?
Power and missing data together determine how many complete, analyzable observations
actually remain — Missing Data shows the other half of that equation.
Significance level α
α0.05
0.010.050.10
Test direction
Design
Sample size
n (per group)64
Total N: 128
Planned effect size
Cohen's d0.50
Power: —
Target power
1−β0.80
📋 Example — clinical antidepressant trial
A researcher is planning a study: expects d = 0.50 (from the literature),
n = 64 per group → planned power:
80%.
A pilot study points to d ≈ 0.30 —
Section 3 shows what that means.
Every conceivable true δ has a different power. The often-cited "80% power" applies only for exactly one assumed δ — and you never know that value for certain before the study.
α and power move in opposite directions
A stricter α (e.g. 0.01 instead of 0.05) lowers power for the same n and δ — you make it harder to reject H₀, and thereby also miss more real effects.
Cohen: rules of thumb, not laws
Cohen (1988) derived d = 0.2/0.5/0.8 from the behavioral-science studies of his time — as a stopgap for when no domain knowledge is available. In medicine, d = 0.1 can be clinically meaningful; in basic research, d = 0.8 can be trivial. The relevant ES is always context-specific.
The MDES (Minimum Detectable Effect Size) should come from clinical/practical relevance, not from statistical conventions. Ask: "What effect would be small enough to ignore?"
Post-hoc power is misleading
Computing power after a non-significant study using the observed d is circular: it just reflects the p-value back at you and says nothing about the original planning. Use only a-priori power (before the study).
Power & Sample Size — Help
Example
A clinician is planning an RCT for an antidepressant. From the literature they expect Cohen's d = 0.50 on the HAM-D scale. With n = 64 per group and α = 0.05 (two-tailed), the planned power comes out to about 80%. A pilot study, however, points to d ≈ 0.30 — Section 3 shows what that means for the actual power.
Power (1−β) is the probability of detecting a true effect — that is, correctly rejecting H₀ when H₁ holds. Power depends on four quantities: effect size δ, sample size n, significance level α, and test direction.
Power = P(reject H₀ | H₁ true) = 1 − β
Why is power a curve?
Every possible true δ yields a different power. At δ = 0, power equals the α level (false positive). As δ grows, power rises toward 1. Section 1 shows this curve — and how it shifts with n. The often-cited "80% power" always applies only for one specific δ.
Normal approximation of power
Power ≈ Φ(|d| · √(n/2) − z_{α/2}) [two-tailed, 2 groups]
Power ≈ Φ(|d| · √n − z_{α/2}) [two-tailed, 1 group]
Φ = standard normal distribution (cumulative), z_{α/2} = quantile for α/2. The approximation is very accurate for n ≥ 20.
Standardized effect size: d = (μ₁ − μ₂) / σ. The small/medium/large (0.2/0.5/0.8) conventions are historical rules of thumb from 1960s–70s behavioral research — not universal standards. Cohen himself warned against adopting them uncritically. In medicine, d = 0.1 can be clinically meaningful (blood-pressure reduction × millions of patients); in basic research, d = 0.8 can be trivial.
SESOI — how do you determine the relevant effect size?
The core problem of sample-size planning isn't the computation — it's the substantive question: What effect do I want to reliably detect?Lakens (2022) suggests anchoring planning on the SESOI — the Smallest Effect Size of Interest: the smallest effect that would still be practically or clinically meaningful.
The SESOI is fundamentally different from the question "what did earlier studies find?" — because published effect sizes are systematically overestimated by selection bias and publication bias (the winner's curse). Anyone who bases their planning on the mean published d risks an inflated effect expectation and therefore chronic underpowering.
Four ways to determine the SESOI:
1. Clinical/practical anchor (recommended): Which effect would change a decision — e.g. treatment guidelines, approval, reimbursement? In medicine, many scales already have empirically determined minimum thresholds (MCID, Minimal Clinically Important Difference). Example: on the HAM-D, a reduction of ≥ 3 points is considered clinically meaningful — which corresponds to a specific d.
2. Resource-based justification: Work backward: what n is realistically achievable? What MDES does that give you at 80% power? Is that MDES still substantively relevant — or so large that only large, implausible effects would be detected?
3. Lower bound from theory: What does theory say about the minimum size of the effect? If a mechanism is postulated, you can often argue below which threshold it practically doesn't exist.
4. Prior literature — with caution: Meta-analytic estimates from pre-selected samples overestimate the true effect. Better: uncorrected raw data from registry studies or preregistered replications. If only published effects are available, set the planning value noticeably more conservatively (e.g. 50–70% of the published d).
Connecting SESOI and MDES: Once the SESOI has been substantively determined, Section 3 of this tool provides the answer: is my planned n large enough to reliably detect the SESOI (MDES ≤ SESOI)? If not, n must be increased or the design changed.
Reference:Lakens, D. (2022). Sample size justification. Collabra: Psychology, 8(1), 33267.
doi:10.1525/collabra.33267 — open access, includes R code and interactive examples. · Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Erlbaum.
Terms
α (Type I error): probability of incorrectly rejecting H₀. β (Type II error): probability of missing a real effect. Power = 1−β. NCP (non-centrality parameter): δ · √(n/2) — indicates how far H₁ is from H₀ (in SD units of the test statistic). MDES (Minimum Detectable Effect Size): smallest effect that can be detected with given n and α at target power. Readable from Section 3. SESOI (Smallest Effect Size of Interest): the smallest effect that would still be substantively practically or clinically relevant. Planning should ensure MDES ≤ SESOI. Post-hoc power: power computed after the study using the observed d — circular and uninformative, because it just reflects the p-value back at you.