Statistical Power and Sample Size
Understand the relationship between α, β, power, and effect size through interactive visualizations. Learn why underpowered studies fail to find real effects.
You'll learn
Why small samples produce unreliable results and how power affects conclusions.
Use this when
You are planning a study and need to justify your required sample size.
The Core Idea: Two Overlapping Distributions
Every hypothesis test involves two distributions: the null (H₀) and the alternative (H₁). The overlap between them determines how hard it is to tell the two apart — which determines power and error rates.
- ●Blue area right of cutoff = Type I error (α) — rejecting H₀ when it is true (false positive)
- ●Amber area left of cutoff = Type II error (β) — failing to reject H₀ when H₁ is true (false negative)
- ●Power = 1 − β = the probability of detecting a real effect
Effect Size: The Signal You Are Trying to Detect
Effect size measures the magnitude of a difference, independent of sample size. Cohen's d is the most common standardized measure for comparing two means:
Cohen's d = (μ₁ − μ₀) / σ
| Cohen's d | Interpretation | Example |
|---|---|---|
| 0.2 | Small | IQ difference between two teaching methods |
| 0.5 | Medium | Drug vs. placebo on a symptom score |
| 0.8 | Large | A highly effective intervention |
| ≥ 1.0 | Very large | Rarely seen in clinical research |
💡 Larger effect = more separation = easier to detect
In the chart above, μ₁ − μ₀ = 2 and σ = 1, so Cohen's d = 2. The distributions barely overlap. With d = 0.2 (small effect), they overlap enormously — you need a much larger sample to detect it reliably.
Sample Size: The Lever You Control
You cannot change the true effect size in nature, and you must set α (usually 0.05). The only lever you control is n. Larger n shrinks the standard error (σ/√n), which pulls the two distributions apart and increases power.
| Cohen's d | n per group for 80% power | n per group for 90% power |
|---|---|---|
| 0.2 (small) | 394 | 527 |
| 0.5 (medium) | 64 | 86 |
| 0.8 (large) | 26 | 34 |
| 1.0 (very large) | 17 | 22 |
⚠️ The most common mistake in clinical research
Underpowered studies are not neutral — they are biased toward false negatives. A study with 40% power that finds 'no significant effect' has a 60% chance of missing a real one. Always power your study before collecting data.
The standard target is 80% power with α = 0.05. This means a 1-in-5 chance of missing a real effect and a 1-in-20 chance of a false positive — a deliberate tradeoff chosen by convention.
Practice with your own dataset
Run a power analysis before designing your study.
- 1.Estimate your expected effect size from the literature
- 2.Open the Power Analysis tool in VibeResearch
- 3.Set α = 0.05 and target power = 0.80
- 4.Adjust effect size to see how n changes — notice the nonlinear relationship
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Further reading
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