Introduction

When studying hypothesis testing, three important concepts appear:

• Alpha error (Type I error)
• Beta error (Type II error)
• Statistical power

These are often introduced through definitions, but to truly understand them, it’s helpful to see them on a graph. In this article, we’ll use normal distributions and interactive visuals to explain:

• What hypothesis testing is
• What α and β errors are
• What statistical power means
• How these relate to areas under curves

Setup: One-Sided Test on a Normal Mean

Let’s consider a simple hypothesis test for the mean of a normal distribution:

• Null hypothesis ($H_0$): $\mu = 0$
• Alternative hypothesis ($H_1$): $\mu = \mu_1 > 0$ (right-sided test)

We assume the test statistic $Z$ follows a standard normal distribution:

Alpha Error: Rejecting a True Null Hypothesis

The alpha error (Type I error) is the probability of rejecting the null hypothesis when it is actually true. It corresponds to the right tail of the standard normal curve under $H_0$:

Here, $z_\alpha$ is the critical value determined by the chosen significance level.

Beta Error: Failing to Reject a False Null Hypothesis

The beta error (Type II error) is the probability of not rejecting $H_0$ when it is actually false (i.e., when $\mu = \mu_1$). This corresponds to the left side of the $H_1$ distribution:

Note that the distribution is now shifted; we’re evaluating the critical threshold under the alternative hypothesis.

Statistical Power

Power is the probability of correctly rejecting $H_0$ when it is false. It is simply:

This is the right tail of the $H_1$ distribution.

Visual Demo

The following interactive demo lets you adjust $\mu_1$, $\sigma$, and the critical value $z_\alpha$ to see how α error, β error, and power are represented as areas under two overlapping normal curves.

In the graph:

• The blue curve represents $H_0$
• The red curve represents $H_1$
• The vertical line is the critical threshold $z_\alpha$
• The blue tail area beyond the threshold = α error
• The red area to the left of the threshold = β error
• The red area to the right of the threshold = power

Summary

• α error: False positive — area under $H_0$ beyond $z_\alpha$
• β error: False negative — area under $H_1$ below $z_\alpha$
• Power: True positive rate — area under $H_1$ beyond $z_\alpha$
• Always pay attention to which distribution each probability is computed under