Calculate Power from a Critical Value
Use a z-based test. Effect and standard error must use the same measurement units.
Example Data
| Test type | Critical z | Expected effect | Standard error | Noncentrality | Estimated power |
|---|---|---|---|---|---|
| Two-tailed | 1.9600 | 0.5000 | 0.2000 | 2.5000 | 70.54% |
| Upper-tailed | 1.6450 | 0.5000 | 0.2000 | 2.5000 | 80.37% |
| Lower-tailed | 1.6450 | -0.5000 | 0.2000 | -2.5000 | 80.37% |
Understand Statistical Power
Statistical power describes the chance of detecting a real effect. How likely is a test to reject the null hypothesis when the alternative is true? Higher power lowers the chance of missing an important result. Researchers often target eighty percent power. Some regulated studies target ninety percent. The best target depends on consequences, cost, and available participants.
A critical value defines the rejection boundary for a test statistic. In normal testing, this boundary is a z value. A larger critical value makes rejection harder. That lowers false-positive risk. It can reduce power when inputs stay fixed. This calculator combines the boundary, expected effect, and standard error.
Why the Expected Effect Matters
The expected effect is the difference assumed under the alternative hypothesis. It may represent a mean, rate, or measured contrast. The calculator divides it by the standard error. This creates the noncentrality parameter. A positive parameter moves the test statistic toward an upper rejection region. A negative parameter moves it toward a lower rejection region. Direction always matters.
Use an expected effect that is scientifically meaningful. Do not choose an extreme value only to obtain high power. Unrealistic assumptions can make a study look stronger than it is. Prior studies and pilot data can guide the estimate. When uncertain, calculate several scenarios. Compare optimistic, central, and conservative effect assumptions.
Formula Used
Let δ equal expected effect divided by standard error. Let Φ represent the normal cumulative distribution. For an upper-tailed test, power equals 1 minus Φ of critical z minus δ. For a lower-tailed test, power equals Φ of negative critical z minus δ. For a two-tailed test, add the rejection probabilities.
The calculator also estimates alpha from the critical value. For a one-sided test, alpha equals 1 minus Φ of the positive critical value. For a two-sided test, alpha equals twice that upper-tail probability. These formulas apply to z tests. Small samples may require a t distribution instead.
How to Use This Calculator
Select the rejection direction first. Choose two-tailed when either direction matters. Choose upper-tailed when a positive effect supports the claim. Choose lower-tailed when a negative effect supports the claim. Enter a positive critical z magnitude. Then enter expected effect in original units. Enter standard error in matching units.
Select Calculate Power. The result box appears before the form. It reports power, beta risk, alpha, and the noncentrality parameter. Check direction against the expected-effect sign. A positive effect in a lower-tailed test produces low power. A negative effect in an upper-tailed test behaves similarly. Revise assumptions and recalculate as needed.
Interpret Results Carefully
Power is a planning measure, not proof of truth. It does not prove a significant result correct. It cannot guarantee useful estimates. Precision, bias control, missing data, and design remain important. Use this result with sample-size calculations and sensitivity checks. Clearly report the important underlying assumptions behind every power estimate.
Frequently Asked Questions
1. What does statistical power mean?
Statistical power is the probability of rejecting the null hypothesis when a specified alternative is true. It measures planned sensitivity to an effect. A power of 80% means the test detects that assumed effect in about eight out of ten comparable studies.
2. Which critical value should I enter?
Enter the positive z magnitude used by your rejection rule. Common values include 1.645 for a one-sided 5% level and 1.960 for a two-sided 5% level. Use the value required by your protocol or statistical plan.
3. Can I enter a negative critical value?
No. Enter a positive magnitude. The selected lower-tailed option applies the negative rejection boundary automatically. This keeps the form clear and prevents accidental sign errors.
4. Does this calculator work for t tests?
It provides a normal-approximation estimate only. Exact t-test power depends on degrees of freedom, the t critical value, and the noncentral t distribution. Use a dedicated t-test power method when samples are small or population variance is unknown.
5. Why is my calculated power low?
Power falls when the expected effect is small, the standard error is large, or the critical value is strict. A direction mismatch can also reduce one-sided power. Consider a larger sample, improved measurement precision, or a realistic design change.
6. What is beta error probability?
Beta is the probability of not rejecting the null hypothesis under the entered alternative scenario. It equals one minus power. It is often described as the chance of a false negative for that specific assumed effect.
7. Why can two-tailed testing lower power?
A two-tailed test divides the rejection risk across both tails. Its positive critical boundary is usually farther from zero than a comparable one-sided boundary. Therefore, it normally needs stronger evidence and may have lower power in one expected direction.
8. What happens when the expected effect is negative?
A negative effect moves the alternative distribution left. It increases power for a lower-tailed test and reduces power for an upper-tailed test. For two-tailed testing, the calculator includes either extreme direction.
9. Can I use a standardized effect?
Yes. Enter a standardized expected effect and its corresponding standard error. The two values must use compatible scales. The calculator only needs their ratio, which becomes the noncentrality parameter.
10. Is 80 percent power always enough?
No. Eighty percent is a common planning convention. Studies with serious safety, policy, or regulatory consequences may require more power. Costs, feasible sample size, expected benefit, and false-negative consequences should guide the target.
11. Does high power guarantee significance?
No. High power improves the long-run chance of detecting the assumed effect. A single study can still miss it because of sampling variation, measurement issues, missing data, or a true effect that differs from the planning assumption.