Non-Central t Distribution
X ~ t_ν(δ)
Noncentral / Heavy-Tailed Family • Published July 22, 2026
The distribution behind t-test power analysis and confidence intervals for Cohen's d: history from Jacob Cohen's power-analysis tradition, a shifted-Normal-over-Chi mixture integral verified against scipy.stats.nct to 3.7e-12, the first t-like sheet in this series where the classic symmetry deliberately breaks (nonzero skewness for δ≠0, with an exact δ=0 tie back to the already-published Student's t), no closed-form maximum-likelihood system, and simulation via its own defining construction.
This reference sheet is part of an Analyze subscription.
Subscribe to read the full Non-Central t Distribution reference sheet, along with every other resource in the library.
See plansPrefer a saved copy?
About This Reference Sheet
The distribution behind t-test power analysis and confidence intervals for Cohen's d: history from Jacob Cohen's power-analysis tradition, a shifted-Normal-over-Chi mixture integral verified against scipy.stats.nct to 3.7e-12, the first t-like sheet in this series where the classic symmetry deliberately breaks (nonzero skewness for δ≠0, with an exact δ=0 tie back to the already-published Student's t), no closed-form maximum-likelihood system, and simulation via its own defining construction.
Support
x ∈ (-∞, ∞)
Parameters
ν > 0 (degrees of freedom), δ ∈ ℝ (noncentrality)