Achieved power for paired means

When to use it

Assess the sensitivity of a before-and-after or matched-pairs comparison of a numerical outcome at the number of pairs available.

Not for

Do not enter the standard deviation of the raw measurements: the test depends on the standard deviation of the within-pair differences.

Notation

$n$: Number of pairs.
$\Delta$: Mean of the within-pair differences.
$\sigma_D$: Standard deviation of the within-pair differences.
$\nu$: Degrees of freedom, n − 1.

Formulas

Effect on the differences and noncentrality

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

A paired t-test is a one-sample t-test on the differences, so the alternative 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

Number of pairs $n$

Values: Integer, at least 2

Enter the complete pairs available for analysis.

Mean difference $\Delta$

Values: Any finite value in outcome units

The expected mean of the within-pair differences.

SD of the differences $\sigma_D$

Values: Positive

From a previous paired study, or from the two SDs and their correlation: σ_D = √(σ₁² + σ₂² − 2ρσ₁σ₂).

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

Pairs are independent of one another, and the within-pair differences are approximately normal.
One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

Worked examples

A worked input set

For 34 pairs, a mean difference of 5, a standard deviation of the differences of 10, α = 0.05, and a two-sided test, use d = 0.5 and 33 degrees of freedom.

Sample size: 34

Open this example in the calculator

Validation

The calculation evaluates the noncentral-t rejection probability directly. It agrees with R’s power.t.test(type = "paired"), and at the size the paired-means calculator returns it reaches that calculator’s target power.

How to cite

StatWisely. Achieved power for paired means. Method document, engine version 1.0.0, reviewed 2026-10-03. https://statwisely.com/methods/power-paired-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.