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    Method document

    Achieved power for two independent means

    Calculates the probability that an equal-size, two-sample t-test rejects its null hypothesis for a specified per-group sample size, mean difference, standard deviation, significance level, and test direction.

    Engine version 1.0.0 · Reviewed 3 October 2026

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    https://statwisely.com/methods/power-two-sample-mean/

    When to use it

    • Assess the sensitivity of a planned equal-allocation comparison of two independent means.

    Not for

    • Unequal allocation, paired observations, and clustered designs require different variance and degrees-of-freedom calculations.

    Notation

    nn
    Participants in each group.
    Δ\Delta
    Difference between group means.
    σ\sigma
    Common population standard deviation.
    ν\nu
    Degrees of freedom, 2n − 2.

    Formulas

    Standardized effect and noncentrality

    d=Δσ,λ=∣d∣n2,ν=2n−2d = \frac{\Delta}{\sigma}, \qquad \lambda = |d|\sqrt{\frac{n}{2}}, \qquad \nu=2n-2

    n is the size of each group. The alternative distribution is a noncentral t with ν degrees of freedom [1].

    Power from the rejection region

    Power=P(Tν,λ>t1−α/tails,ν)+Itails=2P(Tν,λ<−t1−α/tails,ν)\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 per groupnn

    Values: Integer, at least 2

    Enter the participants available in each group.

    Mean differenceΔ\Delta

    Values: Any finite value in outcome units

    Difference between the expected group means.

    Common standard deviationσ\sigma

    Values: Positive

    Assumed equal in both populations.

    Significance levelα\alpha

    Values: Between 0 and 1

    The type I error rate for the specified test.

    Test directionss

    Values: 1 or 2

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

    Assumptions

    • The groups are independent, equally sized, and share the specified population standard deviation.
    • One-sided tests use the upper rejection tail. Two-sided tests split α equally between both tails.

    Worked examples

    A worked input set

    With 30 participants per group, a mean difference of 5, common standard deviation 10, α = 0.05, and a two-sided test, use d = 0.5 and 58 degrees of freedom.

    Group 1: 30 · Group 2: 30 · Total: 60

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    Validation

    • The calculation evaluates the noncentral-t rejection probability directly, using 2n − 2 degrees of freedom.

    How to cite

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

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

    References

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