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The correlation coefficient computed under a conditional distribution, using its conditional covariance and positive conditional variances. It can depend on the observed conditioning value. For a multivariate normal distribution, conditioning on other coordinates gives the corresponding partial correlation.

Treat the displayed entries as the lower triangle of a symmetric sample covariance matrix $S$, in the order mechanics, vectors, algebra, analysis, statistics. Compute the precision matrix $K=S^{-1}$. For a multivariate normal distribution, conditioning on all remaining coordinates givesThese identities follow by fixing the other coordinates in the quadratic form of the normal density and completing the square. For the second identity, the conditional precision matrix for the retained pair is its $2\times 2$ principal block; inverting that block gives the negative off-diagonal entry divided by the square root of the two diagonal entries.

Thus the two quantities to evaluate areAn equivalent procedure uses the Schur complement covariance: regress mechanics on the other four variables, or regress analysis and statistics separately on the remaining three. The first residual variance is $S_{11}-S_{1,-1}{S}_{-1,-1}^{-1}{S}_{-1,1}$; the second result is the partial correlation of the two residuals. No arithmetic is required to specify this method.

The model qualification is important: a covariance matrix alone determines linear-regression residual variances and partial correlations. Identifying them with conditional quantities independent of the observed conditioning values uses the multivariate normal model, or another model with the same conditional moments; it does not follow for arbitrary distributions.

Write ${\sigma }_i^2=V_{ii}$ and $V_{ij}={\rho }_{ij}{\sigma }_i{\sigma }_j$. Apply the Schur complement covariance to the first two coordinates, conditioning on the third. The conditional covariance and conditional variances areDivide the conditional covariance by the product of the conditional standard deviations. The conditional correlation is thereforeThis is the partial correlation after removing the linear effect of $X_3$. For a nonsingular multivariate normal distribution, both residual variances are positive and the answer is independent of $x_3$. If a residual variance vanishes, the requested correlation coefficient is undefined, even though the residual covariance formula remains valid.