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Walk-through

How to Use This Calculator

4 steps

1

Set the effect size, tails, and significance level

Enter the standardized effect size you expect (Cohen's d — 0.2 is small, 0.5 is medium, 0.8 is large), choose one-tailed or two-tailed, and pick your significance level (α), usually 0.05. These three inputs apply to both calculation modes below.

2

Choose Power or Required Sample Size

Use the Power tab when you already know (or plan to use) a sample size and want to know how likely the test is to detect a real effect. Use the Required Sample Size tab when you know the power you want and need to know how many participants per group to recruit.

3

Enter your sample size or target power

On the Power tab, enter the sample size per group. On the Required Sample Size tab, pick a target power — 80% is the conventional default, with 90% used for higher-stakes studies. The calculator recomputes instantly as you type or change a selection.

4

Read the result and interpretation

The result card shows the headline number (power as a percentage, or the required sample size), the Type II error rate (β), and a plain-English interpretation. The Interpretation tab adds Cohen's benchmark table for reading effect sizes in context.

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Reference

Formula & Methodology

2 formulas

Statistical power (from sample size)

power = 1 − Φ(zα − λ) + Φ(−zα − λ), λ = d·√(n / 2)

Φ is the standard normal cumulative distribution function, zα is the critical z-value for your significance level and tails, and λ is the noncentrality parameter for two equal-size groups of n each with standardized effect size d. For a one-tailed test, drop the Φ(−zα − λ) term (it's the opposite-direction rejection region, which a one-tailed test never uses).

Required sample size (from target power)

n = 2 · ((zα + zβ) / d)²

zβ = Φ⁻¹(target power) — for 80% power, zβ ≈ 0.8416; for 90% power, zβ ≈ 1.2816. This is the inverse of the power formula above, solved for n per group. The result is always rounded up, since a fractional participant isn't possible and rounding down would under-power the study.

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Glossary

Key Terms Explained

7 terms

Statistical power The probability that a hypothesis test correctly rejects a false null hypothesis — in other words, the chance of detecting a real effect when one truly exists. Power = 1 − β. Conventionally, researchers aim for at least 80% power.

Type II error (β) A false negative: failing to detect a real effect that actually exists. β is the probability of this happening, and it's the complement of power (β = 1 − power). An underpowered study has a high β — it's likely to miss real effects.

Effect size (Cohen's d) A standardized measure of how large a difference or relationship is, independent of the original units — the difference between two means divided by the pooled standard deviation. Cohen's benchmarks: 0.2 = small, 0.5 = medium, 0.8 = large.

Significance level (α) The probability of a Type I error — rejecting a true null hypothesis (a false positive) — that you're willing to accept. The conventional default is 0.05 (5%). Lower α makes it harder to detect an effect, which lowers power unless you compensate with a larger sample.

Sample size (n) The number of observations per group used in the test. This calculator assumes two independent groups of equal size and asks for n per group, not the combined total across both groups.

1 − β Another name for statistical power — the probability of correctly detecting a true effect. It's written this way because β (the Type II error rate) and power are complementary: together they always sum to 1.

One-tailed vs. two-tailed A two-tailed test splits the significance level across both directions and is the standard, more conservative choice. A one-tailed test puts the entire significance level in one direction, which raises power for a correctly predicted direction but is only valid if you committed to that direction before seeing the data.

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Scenarios

Real-World Examples

3 worked examples

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Planned experiment, power from a known sample size

d = 0.5, α = 0.05 (two-tailed), n = 64 per group

Effect size (d) 0.5 (medium) Significance (α) 0.05, two-tailed Sample size 64 per group

This is the textbook benchmark case: a medium effect size with 64 participants per group at the standard 5% significance level yields roughly 80% power — meaning the test has about a 4-in-5 chance of detecting the effect if it's really there, and about a 1-in-5 chance of missing it (β ≈ 0.20).

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Planning a study, sample size for 80% power

d = 0.5, α = 0.05 (two-tailed), target power = 80%

Effect size (d) 0.5 (medium) Significance (α) 0.05, two-tailed Target power 80%

Working backward from the same 80% target gives a required sample size of about 63-64 participants per group — confirming the well-known rule of thumb that detecting a medium effect at conventional settings takes roughly 64 people per group. Recruiting fewer than this systematically under-powers the study.

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Effect size sensitivity check

d = 0.2 (small) vs. d = 0.8 (large), same n = 64, α = 0.05

Small effect d = 0.2 → power ≈ 25% Large effect d = 0.8 → power ≈ 99%

With the sample size held fixed at 64 per group, power swings dramatically with effect size: a small effect (d = 0.2) is badly underpowered at only about 25%, while a large effect (d = 0.8) is essentially guaranteed to be detected. This is why a realistic estimate of the expected effect size — not just an arbitrary sample size — is the most important input to get right before running a study.

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Deep Dive

Understanding the Statistical Power Calculator

3 chapters · 4 min read

Statistical power answers a question every researcher should ask before collecting data: if the effect I'm looking for is real, how likely is my study to actually find it? Underpowered studies are one of the most common — and most preventable — mistakes in applied research, producing "no significant difference" results that are really just "not enough data to tell." This calculator runs the standard z-approximation power analysis for a two-sample comparison in both directions: power from a planned sample size, or the sample size needed to hit a target power.

How the calculator works

Power depends on four things pulling against each other: the effect size you're trying to detect, your significance level (α), your sample size, and whether the test is one- or two-tailed. Holding the others fixed, power rises as the effect size grows, as the sample size grows, or as α is relaxed (made less strict). This calculator combines those four inputs into a single noncentrality parameter — how far the true effect pushes the sampling distribution away from the null hypothesis — and reads the resulting power off the standard normal curve. It assumes two independent groups of equal size, the most common design in comparative research (A/B tests, treatment-vs-control trials, before/after comparisons with a control group).

Why 80% power is the convention

Jacob Cohen's influential 1988 power-analysis textbook proposed 80% as a reasonable default balance: enough power that a real effect is more likely to be found than missed, without demanding an impractically large sample. That convention stuck, and most journals, grant reviewers, and institutional review boards now expect at least 80% power to be reported or planned for. Some fields — especially clinical trials, where a missed effect can mean an effective treatment gets shelved — push for 90% power instead, accepting the larger required sample size as the cost of a lower false-negative rate.

Limits and edge cases

This calculator uses a z-approximation, which is accurate for reasonably large samples but slightly overstates power for very small samples compared to an exact t-distribution-based calculation (the gap shrinks quickly as n grows past about 20-30 per group). It also assumes equal group sizes, a continuous outcome with roughly normal, equal-variance data, and a between-subjects (independent-groups) design — it is not the right tool for paired/repeated-measures designs, proportions, ANOVA with more than two groups, or regression, each of which has its own power formula. Treat the result as a planning estimate: real-world attrition, unequal group sizes, and violated assumptions all reduce effective power below what the calculator reports.

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Questions

Frequently Asked Questions

6 questions

What is statistical power?+

Statistical power is the probability that a test correctly detects a real effect, if one truly exists. It equals 1 − β, where β is the Type II error rate (the chance of missing a real effect). A power of 80% means an 80% chance of detecting the effect and a 20% chance of missing it.

What target power should I use?+

80% is the conventional default and is widely accepted by journals and reviewers. Higher-stakes research — clinical trials, safety studies, or situations where a missed effect is costly — often targets 90% instead, at the cost of needing a larger sample size.

Does a bigger effect size mean more power?+

Yes. Larger effects are easier to detect, so power rises sharply as effect size increases, holding sample size and significance level fixed. This is also why an unrealistically optimistic effect-size estimate is the most common way studies end up underpowered in practice — the real effect turns out smaller than assumed.

Does a bigger sample size mean more power?+

Yes, and this is usually the variable researchers actually control. Power increases with sample size because a larger sample shrinks the standard error, making a true effect easier to distinguish from random noise. The Required Sample Size tab inverts this relationship to tell you how large a sample you need for a given power target.

What exactly is a Type II error?+

A Type II error means failing to reject the null hypothesis when it's actually false — in plain terms, missing a real effect. It's the complement of power (β = 1 − power). A study with low power has a high risk of Type II error, which is why power analysis matters before, not after, collecting data.

What units does power use?+

Power and the Type II error rate (β) are both probabilities, shown as percentages between 0% and 100%. Effect size (Cohen's d) and significance level (α) are unitless — they don't depend on the original measurement units of your data, which is what makes them comparable across different studies and outcome variables.

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Statistical Power Calculator

Power From Sample Size

Sample Size From Target Power

What Do These Numbers Mean?

How to Use This Calculator

Set the effect size, tails, and significance level

Choose Power or Required Sample Size

Enter your sample size or target power

Read the result and interpretation

Formula & Methodology

Key Terms Explained

Real-World Examples

Planned experiment, power from a known sample size

Planning a study, sample size for 80% power

Effect size sensitivity check

Understanding the Statistical Power Calculator

How the calculator works

Why 80% power is the convention

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

Statistical Power Calculator

Power From Sample Size

Sample Size From Target Power

What Do These Numbers Mean?

How to Use This Calculator

Set the effect size, tails, and significance level

Choose Power or Required Sample Size

Enter your sample size or target power

Read the result and interpretation

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding the Statistical Power Calculator

How the calculator works

Why 80% power is the convention

Limits and edge cases

Frequently Asked Questions

Get a branded PDF of your results

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