200 simulated candidates each get a standardized test score zx and a standardized on-the-job performance score zy, correlated by the test's validity coefficient rxy (a bivariate-normal draw: zy = r·zx + √(1−r²)·noise). The scatter panel plots every candidate directly on the (zx, zy) plane — drag to pan, scroll/pinch to zoom. Sliding "selection ratio" moves a hiring cutoff zc on the test axis; everyone right of it is hired.
Cutoff: P(Z > z_c) = selection ratio (SR)
Success ratio: fraction of hires with z_y ≥ 0 (above-median performer)
Base rate: fraction of ALL applicants with z_y ≥ 0 (what you'd get hiring at random)
Mean z of hires (Taylor-Russell): z̄ = φ(z_c) / SR (φ = standard normal density)
Brogden-Cronbach-Gleser utility:
ΔU = N_hired · T · r_xy · SD_y · z̄ − N_hired · C
N_hired = SR · N_pool(300), T = tenure (yrs), C = $100/applicant tested
- Test validity rxy — how strongly the test predicts real performance. r = 0 means the test is worthless; the cloud becomes a random circle and utility collapses to (near) zero or negative.
- Selection ratio — a tighter cutoff (lower SR) raises the success ratio but hires fewer people, illustrating the Taylor-Russell trade-off. Watch the histogram panel's shaded region shrink as SR tightens.
- SDy and tenure — scale how much one standard deviation of job performance, sustained over the hire's tenure, is actually worth in dollars. Because the utility formula is linear in rxy at fixed SR, the sensitivity panel's utility-vs-validity curve is a straight line through the origin whose slope is set by SR, SDy and tenure together — the marker on it shows exactly where your current test sits on that line.
Real-world relevance: this exact formula is what industrial-organizational psychologists use to justify (or kill) an expensive assessment-center or structured-interview program — a highly valid test still adds little utility if almost everyone gets hired anyway (SR near 100%), and a cheap test can still pay for itself if the selection ratio is tight enough.
Note on the applicant-pool constant: the utility formula is evaluated against a nominal pool of 300 applicants/year (a plausible annual hiring volume), while the scatter/histogram panels sample 200 candidates for visual clarity — the same design choice as the original 3D version, kept here for consistency.