Partial correlation calculator

Researchers, students, and analysts often hold three pairwise correlations from a matrix, a paper, or a spreadsheet and need r_xy.z without rebuilding the raw data. This partial correlation calculator accepts the three pairwise coefficients r_xy, r_xz, and r_yz, applies the classical first-order formula, and returns the partial correlation together with the numerator, denominator, and formula string for audit. When the controlling correlations are perfect in magnitude the denominator is zero and the partial is undefined; those cases fail with invalid_input instead of inventing a number.

What partial correlation measures and the classical formula

Partial correlation of x and y controlling for z, written r_xy.z, is the correlation between the residuals of x on z and the residuals of y on z under linear least squares. When only the three pairwise correlations are available, the same value is obtained from the closed-form identity r_xy.z equals (r_xy minus r_xz times r_yz) divided by the square root of (one minus r_xz squared) times (one minus r_yz squared). This endpoint does not re-estimate coefficients from raw series: it assumes you already have valid pairwise Pearson correlations in the unit interval and focuses on the algebraic step that many worksheets leave to hand calculation. It also does not claim causal control; removing the linear component of z is a statistical adjustment, not proof that z is a confounder or that a causal path has been blocked.

Input contract, response shape, and when the denominator vanishes

POST a JSON body with three required finite numbers, each in the closed interval from minus one to plus one: r_xy (correlation of x and y), r_xz (correlation of x and z), and r_yz (correlation of y and z). Aliases rxy or xy, rxz or xz, and ryz or yz are accepted so compact forms and spreadsheets can reuse the same contract. Missing fields, non-finite values, and values outside the unit interval fail with invalid_input naming the field; those failures are not billed. The critical algebraic guard is the denominator: the product under the square root is zero when the absolute value of r_xz equals one or the absolute value of r_yz equals one, because then x or y is a perfect linear function of z and residual variance collapses. On success the body echoes the three inputs, returns partial_r rounded to twelve decimal places for stable goldens, returns numerator and denominator for audit, and includes a short formula string restating the identity. Example: r_xy equals zero point eight, r_xz equals zero point five, and r_yz equals zero point four yields a positive partial near zero point seven six, showing residual association after adjusting for z.

Where a partial correlation calculator earns its keep

Classrooms introduce partial correlation when contrasting simple association with association after holding a third variable constant, and students need a deterministic oracle that matches the textbook formula on arbitrary three-tuples. Product and data teams sometimes receive only a correlation matrix from a privacy-preserving pipeline and still want first-order partials among selected triples without reconstructing individuals. Keep requests small, assert the rounded partial in tests, and you get reproducible first-order partial correlation for homework, documentation widgets, and edge workers without standing up a full multivariate statistics stack for three numbers. Exam generators and LMS plugins mark items that start from a correlation matrix rather than raw scores; calling this parse endpoint avoids embedding fragile spreadsheet formulas that silently fail when |r_xz| hits one. Social science and epidemiology notes often quote only pairwise r values from prior work and ask for r_xy.z as a sensitivity check before collecting new microdata. Research software tests pin partial_r for known matrix slices so refactors of larger multivariate modules cannot drift.