Achieved power for a one-sample mean

When to use it

Assess the sensitivity of a planned one-sample mean test at the sample size available.

Not for

Do not interpret low achieved power after a non-significant result as evidence that there is no effect.

Notation

$n$: Available participants.
$\Delta$: Mean difference from the null value.
$\sigma$: Assumed population standard deviation.
$\nu$: Degrees of freedom, n − 1.

Formulas

Standardized effect and noncentrality

d = \frac{\Delta}{\sigma}, \qquad \lambda = |d|\sqrt{n}, \qquad \nu=n-1

The alternative distribution is a noncentral t with ν degrees of freedom and noncentrality λ [1].

Power from the rejection region

\text{Power}=P(T_{\nu,\lambda}>t_{1-\alpha/\text{tails},\nu}) + I_{\text{tails}=2}P(T_{\nu,\lambda}<-t_{1-\alpha/\text{tails},\nu})

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Parameters

Sample size $n$

Values: Integer, at least 2

Enter the participants available for analysis.

Mean difference $\Delta$

Values: Any finite value in outcome units

Difference between the expected mean and the null value.

Standard deviation $\sigma$

Values: Positive

Use the standard deviation expected in the study population.

Significance level $\alpha$

Values: Between 0 and 1

The type I error rate for the specified test.

Test direction $s$

Values: 1 or 2

One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Assumptions

The outcome is approximately normal and the standard deviation is the one-sample t-test population SD.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

For 30 participants, a mean difference of 5, standard deviation 10, α = 0.05, and a two-sided test, calculate the noncentral-t probability using d = 0.5 and 29 degrees of freedom.

Sample size: 30

Open this example in the calculator

Validation

The calculation evaluates the noncentral-t rejection probability directly; it does not solve for a required sample size.

How to cite

StatWisely. Achieved power for a one-sample mean. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-one-sample-mean/

Cite the original references for the formulas your design uses as well.

References

1.Cohen J. Statistical Power Analysis for the Behavioral Sciences. 2nd ed. Hillsdale, NJ: Lawrence Erlbaum Associates; 1988.