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Below you will find descriptions and details for the 7 formulas that are used to compute post-hoc statistical power values for Student t-tests.

Beta function:

Cohen's d effect size for a t-test:

where

Gamma function:

Lower incomplete beta function:

Noncentral t-distribution cumulative distribution function:

where

Noncentral t-distribution noncentrality parameter:

where

Regularized lower incomplete beta function:

where the numerator is the lower incomplete beta function, and the denominator is the beta function.

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