What a Power Curve Shows
A power calculation at one assumed effect gives one number. A power curve shows how that number changes across a range of possible true parameter values. It helps answer questions such as: if the true mean were slightly above the null value, how often would this test reject the null hypothesis? What if the true mean were farther away?
A power curve is tied to a particular test and design. Its shape depends on the hypothesis, significance level, sample size, variability, and the model used to calculate power. It is not a plot of observed data and does not show which parameter value is most likely.
The horizontal coordinate is a possible truth about the population, not an estimate calculated from the sample. The vertical coordinate is a probability between 0 and 1. For example, a curve height of 0.80 at a true mean of 52 outcome units means that, if the specified test and study design were repeated many times under a population mean of 52, about 80% of those tests would reject \(H_0\).
Always identify what is held fixed. Unless stated otherwise, a power curve represents one specified test, with a particular sample size, significance level, variability assumption, and direction of testing. A different design can produce a different curve.
Reading the Null and Alternative Values
The value specified by the null hypothesis is a key reference point on the horizontal axis. For a test with exact significance level \(\alpha\), the curve’s height at the null value is \(\alpha\): when the null is true, the probability of rejecting it is the Type I error rate. A curve may show the null value with a vertical reference line or a marked point; if it does not, locate the null value from the hypothesis itself.
At a target alternative, read vertically from that true parameter value to the curve and then horizontally to the power axis. That height gives power at the target. The quantity \(1-\text{power}\) is the probability of failing to reject \(H_0\) at that same alternative, also called the Type II error probability there.
For an upper-tailed test, power usually increases as the true parameter moves above the null value: larger departures in the direction being tested are easier for the test to detect. For a lower-tailed test, the corresponding increase is in the lower direction. For a two-sided test, power can increase in either direction away from the null, so its curve often has a low point near the null and rises toward both extremes. The direction and shape therefore depend on the test, not just on the parameter.
Worked Example: Interpreting a One-Sided Power Curve
Suppose a hypothetical study tests whether the population mean change in a clinical measurement exceeds 50 mmHg. The planned one-sample z test has \(H_0:\mu=50\), \(H_a:\mu>50\), \(n=100\), known population standard deviation \(\sigma=10\) mmHg, and \(\alpha=0.05\). Assume independent observations and a normal population distribution, so the sample mean has a normal sampling distribution and the z test is appropriate.
For this test, the standard error is \(SE=\sigma/\sqrt{n}=10/\sqrt{100}=1\) mmHg. The test rejects for a sufficiently large standardized sample mean. Using the one-sided critical value \(z_{0.95}\approx1.645\), the test rejects when \(\bar{x}>50+1.645(1)=51.645\) mmHg.
To find the curve height at a true mean of 52 mmHg, calculate the chance that the sample mean exceeds 51.645 when the population mean is 52. Under that true mean, \(\bar{x}\) has mean 52 and standard deviation 1. Thus \[ P(\bar{x}>51.645\mid\mu=52) =P\left(Z>\frac{51.645-52}{1}\right) =P(Z>-0.355) =\Phi(0.355)\approx0.6387. \]
So the power curve is about 0.6387 at \(\mu=52\) mmHg. In repeated studies conducted under this true mean and the stated assumptions, the test would reject \(H_0\) about 63.87% of the time. This is not a statement that there is a 63.87% probability that the true mean is 52 mmHg.
At the null value \(\mu=50\) mmHg, the curve is at 0.05, the test’s significance level. At larger means in the tested direction, the curve rises; for instance, it is about 0.9123 at \(\mu=53\) mmHg. These points describe the same fixed design, not different observed results.
Why the Curve Has This Shape
The one-sided z test in the example also gives a compact formula for the curve. Suppose the null value is \(\mu_0\), the alternative is \(\mu>\mu_0\), the population standard deviation \(\sigma\) is known, and the observations are independent and normally distributed. Then \(\bar{x}\) has mean \(\mu\) and standard error \(\sigma/\sqrt{n}\). The test rejects when its standardized statistic exceeds \(z_{1-\alpha}\).
Under a particular true mean \(\mu\), the standardized statistic has a normal distribution with variance 1 and mean \((\mu-\mu_0)/(\sigma/\sqrt{n})\). The test’s power is therefore the probability that this statistic exceeds the rejection cutoff. This gives the curve as a function of the true mean:
Here \(\Phi\) is the standard normal cumulative distribution function. At \(\mu=\mu_0\), the expression becomes \(\Phi(-z_{1-\alpha})=\alpha\). As \(\mu\) increases above \(\mu_0\), the standardized distance \((\mu-\mu_0)/(\sigma/\sqrt{n})\) increases, so the power increases. This explains both the null-point height and the upward shape for an upper-tailed test.
This formula is specific to the stated z-test model and assumptions. Other tests can have different formulas and curves, but the basic reading remains the same: the curve height at a true parameter value is the probability of rejection under that value.
How Sample Size Changes the Curve
For a fixed \(\alpha\), null value, variability, and target alternative in the tested direction, increasing \(n\) reduces the standard error \(\sigma/\sqrt{n}\). The target alternative is then farther from the null in standard-error units. The test can distinguish the target from the null more clearly, so power increases. At the null point, however, the curve remains at \(\alpha\): increasing sample size does not change the test’s Type I error rate when \(\alpha\) is held fixed.
Worked Example: Comparing Curves for Different Sample Sizes
Consider the same hypothetical upper-tailed test of \(H_0:\mu=50\) mmHg against \(H_a:\mu>50\), with known \(\sigma=10\) mmHg and \(\alpha=0.05\). Compare planned sample sizes \(n=25\), \(n=100\), and \(n=400\) at the target \(\mu=52\) mmHg. Assume independent, normally distributed observations for each design.
The critical value is \(z_{0.95}\approx1.645\). For \(n=25\), \(SE=10/\sqrt{25}=2\) mmHg, so the standardized distance from the null to the target is \((52-50)/2=1\). The power is \(\Phi(1-1.645)=\Phi(-0.645)\approx0.2595\).
For \(n=100\), \(SE=10/\sqrt{100}=1\) mmHg. The standardized distance is \((52-50)/1=2\), giving power \(\Phi(2-1.645)=\Phi(0.355)\approx0.6387\).
For \(n=400\), \(SE=10/\sqrt{400}=0.5\) mmHg. The standardized distance is \((52-50)/0.5=4\), giving power \(\Phi(4-1.645)=\Phi(2.355)\approx0.9907\).
At this target, the three designs have powers of about 0.2595, 0.6387, and 0.9907, respectively. On a graph, the larger-sample curves would have greater height at \(\mu=52\) mmHg and at other alternatives above 50 mmHg. All three curves still have height 0.05 at the null value, because \(\alpha\) is unchanged.
How Significance Level Changes the Curve
For a fixed sample size and model, increasing \(\alpha\) makes the rejection rule less stringent. In an upper-tailed z test, the critical value \(z_{1-\alpha}\) becomes smaller, so rejection becomes more likely under alternatives as well as under the null. Consequently, the curve is higher at alternatives, but its height at the null also rises to the new \(\alpha\). Decreasing \(\alpha\) has the reverse effect: lower power at alternatives, in exchange for a lower Type I error rate.
Worked Example: Comparing Curves at Different Significance Levels
Return to the hypothetical upper-tailed test with \(\mu_0=50\) mmHg, \(n=100\), and known \(\sigma=10\) mmHg. The standard error is 1 mmHg. Compare \(\alpha=0.01\), \(0.05\), and \(0.10\) at the same target, \(\mu=52\) mmHg. The same independence and normality assumptions apply.
At \(\alpha=0.01\), \(z_{0.99}\approx2.326\). The target is 2 standard errors above the null, so power is \(\Phi(2-2.326)=\Phi(-0.326)\approx0.3722\).
At \(\alpha=0.05\), \(z_{0.95}\approx1.645\), giving \(\Phi(2-1.645)=\Phi(0.355)\approx0.6387\).
At \(\alpha=0.10\), \(z_{0.90}\approx1.282\). Using that displayed rounded critical value, power is \(\Phi(2-1.282)=\Phi(0.718)\approx0.7636\).
Thus, at \(\mu=52\) mmHg, the higher significance levels give higher power: about 0.3722, 0.6387, and 0.7636. But at the null value, the curve heights are 0.01, 0.05, and 0.10, respectively. The curves do not simply shift upward without consequence; increasing power this way also permits a higher Type I error rate.
A Practical Reading Checklist
When given a power curve, read it in a consistent order. First identify the parameter and its units on the horizontal axis, then identify power on the vertical axis. Next locate the null value from \(H_0\), and check that the curve’s height there agrees with the stated \(\alpha\) for an exact-level test. Finally, locate the target alternative and read its power. If curves are compared, check which design feature differs before explaining why one curve is higher.
Name the population parameter on the horizontal axis and interpret the vertical scale as rejection probability under each true parameter value.
Use \(H_0\) to identify its horizontal location. For an exact-level test, the curve height at that point is \(\alpha\).
At the target alternative, move up to the curve and across to the power axis. State the probability in context.
State which feature changed, such as sample size or \(\alpha\), and relate the observed curve difference to that change.
Common Mistake: Treating Power as Evidence About the Truth
A common mistake is to say, “The power is 80%, so there is an 80% chance the effect is real.” That reverses the conditioning. Power is calculated assuming a particular true parameter value and describes the probability that the test rejects \(H_0\) under that assumption. It is not the probability that the assumed value is true given a test result. Similarly, a high power curve does not guarantee a significant result in one particular study; it describes a long-run probability across repeated studies under specified conditions.
Another useful caution is that the power curve does not, by itself, indicate whether an effect is practically important. The horizontal axis gives parameter values, so interpret distances and units in the scientific context. A high power at a very small difference says the test can often detect that difference under its assumptions; it does not determine whether that difference matters in practice.
Check Your Understanding
Use the definitions and interpretation rules in this tutorial to answer the following questions.
- On a power curve for a test of \(H_0:\mu=20\), what does the height at \(\mu=20\) represent when the test has exact significance level 0.05?
- A curve has height 0.72 at a specified alternative. Explain what 0.72 means in context, and state what \(1-0.72\) represents.
- For an upper-tailed test with \(\alpha\) held fixed, how should increasing sample size affect power at a target above the null? What happens to the curve height at the null?
- For a fixed sample size, what happens to the null-point height and alternative power when \(\alpha\) increases?
- Why is it incorrect to interpret power at a target value as the probability that the target value is true?