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Power Analysis Calculator

Estimate statistical power (1 − β), required sample size, minimum detectable effect, and Type II error for supported study designs. Choose design, outcome, α, and effect—or reverse-solve for power, n, or MDE from existing parameters.

What do you want to calculate?

Study design

Hypothesis

Test direction

Statistical power = 1 − β = probability of detecting a true effect of the specified size at the chosen significance level α.
Power calculations are only as reliable as the assumed effect, variability, allocation, and design. They are not a guarantee of study success or a substitute for a statistical analysis plan.
For general survey estimation sample size (confidence, margin of error, FPC), use the Sample Size Calculator.
Statistical power
80.74%

With the specified sample size, effect size, significance level, and two-sided test, the estimated statistical power is 80.74%. Under the assumed alternative effect, the study would have approximately a 80.74% probability of detecting that effect at the chosen significance level.

Method
two_means power analysis
Power (1 − β)
80.74%
Type II error β
19.26%
Type I error α
5.00%
Sample size (total)
128
Group 1 n
64
Group 2 n
64
Effect / difference used
—
MDE
—
Cohen's d (std. effect)
0.5000
Critical z
1.9600
Recruitment target
—

Sensitivity: sample size → power

n (approx total)Powerβ

Power curve (sample size → power)

Dashed lines mark 80% and 90% power. Curve uses current design, effect, and α.

Sensitivity: α → power

αPowerβ

Sensitivity: effect size → power

EffectPower

Assumptions for this design

Calculation steps

  1. Default: two independent means, Cohen's d = 0.5, n₁ = 64, n₂ = 64, α = 5% two-sided
  2. se = √(1/n₁ + 1/n₂) = 0.176777
  3. nc = d / se = 2.828427
  4. zα = 1.959964
  5. Power = (1 − Φ(zα − nc)) + Φ(−zα − nc) = 80.743051%
  6. β = 1 − power = 19.256949%
Post-hoc power based on the observed effect can be misleading. Prefer confidence intervals and effect estimates after a study is complete.

How to Use the Power Analysis Calculator

Choose what to calculate, select the study design, enter the expected effect and variability, set α and test direction, and review the primary result, sensitivity tables, and power curve.

  1. 1. Choose what to calculate: Pick Power to estimate statistical power from a given sample size, Required Sample Size to solve for n given a target power, Minimum Detectable Effect to find the smallest effect detectable at a given n, Type II error (β) to compute the complement of power, or Significance level (α) to reverse-solve the α that achieves a target power at a fixed n and effect.
  2. 2. Select the study design: Choose One-sample mean, Two independent means, Paired means, One-sample proportion, Two independent proportions, Paired binary (McNemar), Pearson correlation, or Multiple linear regression. Each design uses the appropriate statistical engine — noncentral normal, noncentral F, Fisher z, or the Connor/McNemar formula — not a single generic approximation.
  3. 3. Enter the expected effect and variability: For means designs, enter the expected mean difference and SD — or switch to Cohen's d if you already have a standardized effect. For proportions, enter the expected baseline and comparison rates. For correlation, enter the expected r. For regression, enter the expected R², the number of tested predictors, and the total predictors. Variability is what determines how much of the effect is signal versus noise.
  4. 4. Choose α, hypothesis, and test direction: Set the significance level (default 0.05). Choose the hypothesis framework: Superiority, Non-inferiority, or Equivalence (TOST). If superiority, choose Two-sided or One-sided testing — one-sided increases power but should only be used when the direction is prespecified. For non-inferiority or equivalence, a margin must be entered.
  5. 5. Enter sample size or target power: If solving for power, MDE, or β, enter the sample size (per group and total for two-group designs, or the allocation ratio). If solving for required n, enter the target power (80% is a common default). The calculator will solve for the missing quantity.
  6. 6. Review the primary result and β: The hero figure shows the primary output for the selected solve mode. Below it, the stats panel reports power, β, α, per-group and total sample size, effect used, critical z, and recruitment target if attrition is set. A short interpretation line explains what the number means in context.
  7. 7. Review sensitivity tables and the power curve: Three sensitivity tables show how power changes with sample size, with α, and with effect size — so you can see how sensitive the result is to each assumption. The power curve plots power against sample size and marks the 80% and 90% reference lines.
  8. 8. Plan recruitment separately from statistical n: If you expect dropout or attrition, open the Advanced panel and enter the expected rate. The calculator returns the recruitment target separately — statistical n ÷ (1 − dropout rate) — and never changes the statistical n used for power. Keep the two quantities distinct in your protocol.

What Is Statistical Power?

Statistical power is the probability of detecting a true effect of a specified size at a chosen significance level. It equals 1 − β, where β is the Type II error rate — the probability of failing to detect an effect that is actually there. Power of 80% means that, if the true effect is exactly as assumed, the study has an 80% chance of producing a significant result.

Two errors are in play. A Type I error (α) is rejecting H₀ when it is true — a false positive. A Type II error (β) is failing to reject H₀ when a real effect exists — a false negative. Power is the complement of the second. Power is not the probability that a hypothesis is true, and a high-power study does not guarantee a significant result on any single run.

What Power Analysis Does

Power analysis connects the four quantities that determine whether a study can detect an effect: sample size, effect size, α, and power (with β as its complement). Given any three, the fourth is determined. This calculator solves for power, required sample size, minimum detectable effect, or β — for a chosen design and test direction.

It also runs sensitivity analyses: how power changes as sample size grows, as α changes, and as the assumed effect size varies. Those tables and the power curve show how sensitive the conclusion is to each input.

Power Analysis vs Sample Size Calculator

A general sample size calculator answers a survey estimation question: how many observations do I need for a confidence interval with a specified margin of error? A power analysis calculator answers a study design question: how many observations do I need to have a reasonable chance of detecting an effect of interest? The two overlap when a survey is being designed to detect a difference, but they are separate planning problems. For general estimation sample size, use the Sample Size Calculator. For Slovin's finite-population formula, use the Slovin's Formula Calculator.

How Power Is Calculated

There is no single formula that covers every design. The calculator applies the correct method for the selected design:

  • Means (one-sample, two-sample, paired) — noncentral normal approximation. The noncentrality parameter is nc = δ ÷ SE, where δ is the effect and SE is the standard error for that design.
  • Proportions — normal approximation to the binomial, using the null and alternative variances. Paired binary uses the Connor/McNemar formula on discordant pairs.
  • Pearson correlation — Fisher z transform: nc = z(r) × √(n − 3).
  • Multiple regression — exact noncentral F with λ = f² × n, where f² = R² ÷ (1 − R²). Power is computed as 1 − noncentralF_CDF(F_crit; u, v, λ).

For two-sample means with unequal SDs, the calculator uses a Welch-style SE: √(s₁²/n₁ + s₂²/n₂). For equal SDs, the pooled SE √(1/n₁ + 1/n₂) × σ applies.

Required Sample Size

For a target power, the workflow is: expected effect → variability → α → target power → required n. The calculator solves for n and always rounds up — a fractional sample is meaningless. For two-group designs, n is reported per group and in total, and unequal allocation is supported via the n₂/n₁ ratio.

Required n is a planning target, not a guarantee. It depends on the assumed effect size and variability. If the true effect is smaller than assumed, or the true variability is larger, the achieved power will be lower than planned.

Minimum Detectable Effect

The MDE is the smallest effect a study can reasonably detect, given its sample size, α, and target power. It answers a different question than "what effect do I expect?" — MDE is a property of the design, not of the underlying phenomenon. A study with a large MDE can only reliably detect large effects; a study with a small MDE can detect smaller ones but requires more observations. The observed effect after the study is a third quantity that depends on the data.

Effect Size Metrics by Design

  • Cohen's d — standardized mean difference for one-sample, two-sample, and paired means. The calculator can also report Hedges' g, the small-sample-corrected version.
  • Proportion difference — absolute difference (p₂ − p₁) for one- and two-proportion designs. The engine uses the absolute difference even when a relative label is chosen for reporting.
  • Correlation r — Pearson correlation, analyzed via Fisher z.
  • R² and f² — for multiple regression, in overall-model or incremental (added predictors) mode.
  • Discordant proportions — p₀₁ and p₁₀ for paired binary (McNemar).

These are not interchangeable. A d of 0.5 is not the same as r of 0.5, and a 5-percentage-point proportion difference depends on the baseline rate. Choose the metric that matches the design.

Alpha, Beta, and the Trade-Off

Stricter α (e.g. 0.01 instead of 0.05) reduces false positives but lowers power for the same n and effect — you need a larger sample to compensate. Looser α raises power but increases the false-positive rate. Power analysis makes that trade-off explicit: you can see exactly how much n is required to keep power at 80% if you tighten α.

One-Sided vs Two-Sided Testing

One-sided tests have more power for the same n when the direction of the effect is genuinely known in advance. They should be chosen because the scientific hypothesis is directional, not because the sample size requirement is smaller. Switching from two-sided to one-sided after seeing the data is a form of p-hacking. The calculator allows both; the assumptions panel reflects the choice.

A Priori vs Post-Hoc Power

A priori power analysis is performed before data collection to plan sample size. Post-hoc power analysis is performed after the study, using the observed effect. The calculator supports both, but the post-hoc case is caveated: observed-effect power is a deterministic function of the p-value and adds no new information beyond the test statistic — a study that just reached significance will show post-hoc power near 50%, and one that fell short will show low post-hoc power regardless of the true effect. Confidence intervals and effect estimates are the better follow-up.

Attrition: Recruitment vs Statistical Sample

Two quantities must stay separate. The statistical required sample is what the power calculation returns. The recruitment target is that number divided by the expected retention rate. If you expect 20% dropout, you need to recruit required_n ÷ 0.80. The calculator shows both — attrition adjusts only the recruitment target and never the statistical n used for power.

Assumptions and Limits

Power calculations depend entirely on the assumptions entered. The effect size, variability, α, and design all have to be specified; if any is wrong, the power estimate is wrong in proportion. Common assumptions the calculator relies on:

  • Independent observations within each group.
  • Approximately normal residuals or sufficiently large n (for means).
  • Normal approximation to the binomial with reasonable n and proportions not extremely near 0 or 1 (for proportions).
  • Bivariate normality or large-n Fisher z validity (for correlation).
  • Linear model with independent errors (for regression).

A power calculation is not a guarantee of study success. It is a planning estimate that says: under the stated assumptions, this is the probability of a significant result if the effect is exactly as assumed. Real studies rarely match their assumptions precisely.

Common Mistakes

  • Using an inflated effect size. Planning for an unrealistically large effect produces an underpowered study when the true effect is smaller.
  • Ignoring variability. Underestimating SD has the same effect as overestimating the mean difference.
  • Reporting post-hoc power as evidence. It is not independent evidence — it is a p-value restated.
  • Switching from two-sided to one-sided to save sample size. The test direction must be prespecified.
  • Forgetting attrition. Statistical n is not the recruitment target.
  • Treating one design's effect size as another's. d, r, and proportion differences are not interchangeable.
  • Planning for 80% power and treating it as certain. 80% power means a 20% chance of a Type II error under the assumed effect.

Frequently Asked Questions

Q: What is a good statistical power?

80% is the conventional minimum in most fields, and 90% is common for clinical trials and high-stakes studies. There is no universal rule — the appropriate power depends on the cost of a false negative in the specific context. Higher power always requires more observations or a larger effect.

Q: What is the difference between power and sample size?

Sample size is the number of observations. Power is the probability of detecting an effect given that sample size, the effect size, α, and the design. They are two sides of the same equation — given any three of {n, effect, α, power}, the fourth is determined. This calculator solves for any of them.

Q: How does effect size affect power?

Larger effects are easier to detect, so power increases with effect size for a fixed sample size. Conversely, a smaller effect requires a larger sample to maintain the same power. The effect-size sensitivity table shows the exact relationship for the current design.

Q: What sample size do I need for 80% power?

It depends entirely on the effect size, variability, α, and design. There is no universal number. Enter the expected effect, SD, α, and 80% as the target power in the Required Sample Size mode, and the calculator returns the required n per group and in total.

Q: How does unequal allocation affect power?

For a fixed total sample size, a 1:1 allocation maximizes power. Unequal allocation (for example 1:2 or 1:3) reduces power slightly for the same total n because the effective information per participant is lower. The calculator returns per-group and total sample size for the ratio you specify, so you can see the effect directly.

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