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Fix centered standardized moments for odd orders - #179
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The centered algorithm used integer division in N / 2, omitting one factor of the population standard deviation for odd orders. For [0, 0, 1], this returned a third standardized moment of 1/3 instead of 1/sqrt(2). Divide by the missing standard-deviation factor in both overloads while preserving even-order calculations. Correct the centered formula documentation and describe the algorithm parameters. Cover third, fourth, and fifth moments with independent expected values across float and double inputs, range and variadic calls, and positive rescaling. AI-Assisted-By: Codex (GPT-6 Astra Medium)
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The centered standardized-moment algorithm uses integer division in
N / 2, omitting one factor of the population standard deviation for odd orders. For[0, 0, 1], the third standardized moment consequently returns1/3instead of1/sqrt(2).This change divides by the missing standard-deviation factor in both the range and variadic overloads. Even-order calculations, the first- and second-order special cases, and the default scaled algorithm remain unchanged. It also corrects the centered formula documentation and documents the algorithm parameters.
Regression tests cover third, fourth, and fifth moments using independently derived expectations, float and double inputs, both overloads, and positive rescaling.
Validation:
The centered algorithm retains its existing susceptibility to overflow and underflow when computing unscaled central moments and variance powers. This change fixes normalization without redesigning numerical stability.
AI-Assisted-By: Codex (GPT-6 Astra Medium)