What Is Statistical Power?

Statistical power is the probability that a test correctly rejects the null hypothesis when the alternative is true—that is, the chance of detecting an actual effect. It answers the question:

“If there truly is a difference, how likely is the test to detect it?”

A high power means a low risk of Type II error (false negative). Power depends on four factors:

• Significance level $\alpha$
• Effect size $\delta = \mu_A - \mu_0$
• Population standard deviation $\sigma$
• Sample size $n$

This article focuses on how increasing the sample size $n$ improves power and reshapes the power curve.

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The Power Function: A Step-by-Step Derivation

We consider a one-sided $Z$-test with the following setup:

• Null hypothesis: $\mu = \mu_0$
• Alternative hypothesis: $\mu > \mu_0$
• Known standard deviation $\sigma$
• Sample mean $\bar{X} \sim \mathcal{N}(\mu, \sigma^2 / n)$

1. The Test Statistic

The standardized test statistic is:

2. Rejection Region

We reject the null hypothesis if:

where $z_\alpha$ is the upper $\alpha$ quantile of the standard normal distribution, i.e.,

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Noncentral Distribution Under the Alternative

Now suppose the alternative is true, i.e., $\mu = \mu_A$. Then:

So, under the alternative hypothesis:

The mean shift is called the noncentrality parameter:

This tells us the test statistic follows a noncentral normal distribution when the null is false.

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Power Function

Power is the probability that this noncentral distribution falls into the rejection region:

where $\Phi(\cdot)$ is the standard normal cumulative distribution function.

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Example: How Power Changes with Sample Size

Let’s fix parameters as follows:

• Effect size $\delta = 0.5$
• Standard deviation $\sigma = 1$
• Significance level $\alpha = 0.05$

Then compute power for different $n$:

Sample Size $n$	Power
10	0.158
30	0.385
50	0.553
100	0.809

As $n$ increases, the noncentral distribution shifts right, increasing the chance it exceeds $z_\alpha$, and hence increasing power.

Key Takeaways

• Power increases as sample size $n$ increases.
• Under the alternative, the test statistic shifts right → higher chance to reject $H_0$.
• The shift is quantified by the noncentrality parameter $\lambda = \frac{\delta \sqrt{n}}{\sigma}$.
• The power function shows how likely we are to detect a true effect.
• Visualizing power curves helps in planning sample size effectively.