For a W x N null data matrix with population covariance Sigma = I, sample correlation and sample covariance matrices share the Marchenko-Pastur limiting spectral distribution, but they differ at the level of finite-sample fluctuations. This note compares the two null ensembles through the power-sum statistic p2 = tr (Sigmaₕat²), and derives closed-form finite- (N, W) expressions for its first three cumulants in the correlation and covariance cases. The fixed unit diagonal of the sample correlation matrix removes fluctuation sectors that remain present for sample covariance matrices. At third order, this produces a topology-level cancellation: open index topologies vanish in the correlation ensemble, leaving the triangle contribution as the surviving cross-term. Applying the delta method to rₑff = N² / p2 gives a dimensionless sign criterion for effective-rank skewness. In the fixed-aspect-ratio limit gamma = N/W > 0, the criterion yields opposite signs: the effective-rank skewness is negative for the correlation ensemble and positive for the covariance ensemble. Simulations agree with the finite-dimensional cumulant formulas.
Jesus David Calderas Cervantes (Sat,) studied this question.