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). Sliding "selection ratio" moves a hiring cutoff zc on the test axis; everyone above 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_applicants, 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.
- SDy and tenure — scale how much one standard deviation of job performance, sustained over the hire's tenure, is actually worth in dollars.
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.