If the true effect size is small and the sample size is fixed, what happens to the statistical power?

• A. Power decreases.
• B. Power increases.
• C. Power remains constant.
• D. Power becomes unreliable.

Answer: (A)

Power is the probability of detecting an effect when it truly exists, and it depends on the size of that effect, the amount of data, and how noisy the measurements are. When the true effect size is small and the sample size is fixed, the difference the test has to detect is subtle compared to the natural variability in the data. With the same amount of data, the standard error doesn’t shrink, so the test statistic under the alternative isn’t much larger than under the null. That makes it less likely to cross the threshold for significance, so the chance of rejecting the null is lower. In short, smaller true effects are harder to detect without increasing the sample size; power decreases as effect size shrinks when n is fixed. If you wanted more power in this situation, you'd typically increase the sample size (or reduce variability) rather than keep n fixed.