Type II Error Calculator Formula

Understand the math behind the type ii error calculator. Each variable explained with a worked example.

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Formulas Used

Z for Power Calculation

z_power = z_beta

Standard Error

standard_error = se

Non-centrality Parameter

noncentrality = (mu1 - mu0) / se

Effect Size d

effect_size = (mu1 - mu0) / sigma

Variables

Variable	Description	Default
`mu0`	Null Hypothesis Mean	100
`mu1`	True Mean	105
`sigma`	Population SD	15
`n`	Sample Size	25
`z_crit`	Critical z (e.g., 1.96)	1.96
`se`	Derived value`= sigma / sqrt(n)`	calculated
`z_beta`	Derived value`= z_crit - (mu1 - mu0) / se`	calculated

How It Works

Understanding Type II Error

Concept

Beta = P(fail to reject H0 | H0 is false)

Power = 1 - Beta

Type II error occurs when you fail to detect a real effect. The probability depends on the true effect size, sample size, significance level, and population variability. A negative z_beta value indicates high power (likely to detect the effect).

Worked Example

H0: mu = 100. True mu = 105. SD = 15, n = 25, z_crit = 1.96.

mu0 = 100mu1 = 105sigma = 15n = 25z_crit = 1.96

01SE = 15 / sqrt(25) = 15 / 5 = 3
02Non-centrality = (105 - 100) / 3 = 1.667
03z_beta = 1.96 - 1.667 = 0.293
04A z_beta of 0.293 corresponds to roughly beta = 0.615
05Power ≈ 1 - 0.615 = 0.385 (about 39%)
06This sample size gives low power to detect this effect

Frequently Asked Questions

How do I reduce Type II error?

Increase sample size, increase the significance level (accept higher Type I error), reduce measurement variability, or study a larger effect. Sample size is the most practical lever.

What is the relationship between alpha and beta?

For a fixed sample size and effect, decreasing alpha (stricter threshold) increases beta (more Type II errors). There is a tradeoff: reducing one type of error increases the other unless you also increase n.

What is an acceptable beta level?

Conventionally, beta = 0.20 (power = 0.80) is the minimum acceptable level. Clinical trials often aim for beta = 0.10 (power = 0.90) for more reliable detection.

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