Statistical Power & Type I/II Errors (2D)

Test Setup

Sample size n 30

Effect size (Cohen's d) 0.80

Significance level α 0.050

Drag the yellow critical line directly in the top chart to fine-tune α continuously — the select box snaps back to the nearest preset when you let go.

Live Readouts

Critical value z_crit

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Power (1−β)

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Type I error α

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Type II error β

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How it works

Testing H₀: μ = μ₀ against H₁: μ = μ₀ + d (one-tailed, upper rejection region) with a known population SD, the sample mean has a normal sampling distribution under each hypothesis, with standard error:

SE = σ / √n   (σ = 1 assumed)

The critical value is set from α using the standard-normal quantile z_1−α, then converted back into sample-mean units:

x_crit = z(1−α) · SE
Reject H0 when sample mean > x_crit

Because the alternative distribution is just the null shifted by d, its probability mass past x_crit is the statistical power, and the mass that falls short is β (a miss):

Power = 1 − Φ( z(1−α) − d·√n )
β      = Φ( z(1−α) − d·√n )

Teal curve (top chart) — the null sampling distribution. The red fill past the critical line is α, the chance of a false positive (Type I error) even though H₀ is true.
Orange curve — the alternative sampling distribution, shifted right by d·√n standard errors. The blue fill below the critical line is β, a miss (Type II error); the green fill is the power — correctly detecting a real effect.
n slider — a larger sample shrinks SE, narrowing both curves and pulling them apart in z-units, which is why bigger studies detect smaller effects reliably. Watch the left mini-chart trace this out as a power curve against n.
d slider — the true size of the effect being tested for; larger d separates the two curves and raises power directly, traced live in the right mini-chart.
α select / drag — moving the critical line left (larger α) trades a higher false-positive rate for higher power — the classic α/power trade-off behind every "underpowered study" warning.

H₀ sampling distribution H₁ sampling distribution α — Type I error β — Type II error Power (1−β)