Range Restriction — Variance Restriction & Correlation Attenuation

Does: shows how selecting on the predictor X systematically biases the correlation r observed in the selected group — and how Thorndike's formulas back-correct the true value ρ. Does NOT: no estimation from real data, no indirect case with a measured third variable (only a demonstration of the mechanism under bivariate normality).

If a sample covers only part of the value range of X (e.g. because only applicants above a test cutoff are hired), the spread of X shrinks. The correlation between X and Y is then underestimated — even though the true relationship in the overall population is much stronger. This is direct range restriction (Thorndike Case II).

A company validates an aptitude test X against later career success Y. But: only those who score above the cutoff are hired — only for them is a Y value available later. The validity r observed this way is smaller than the true validity ρ. Anyone who ignores range restriction wrongly writes off a good test as useless.

Correlation = standardized covariance. Selecting on X caps the spread of X (s_x drops), while the residual spread of Y around the regression line stays the same. As a result, the systematic variation makes up a smaller share of Y's total variance → r drops. The key quantity is u = s'x / sx (selected spread ÷ total spread).

Under direct selection on X, the regression slope b_Y·X theoretically stays unchanged — only the correlation (and R²) drop. In the right plot, the red line has essentially the same slope as the blue population line; the points just scatter over a narrower X range. Correlation ≠ slope.

Top left: overall population. Gray points = not selected, red = selected. The orange line is the selection cutoff on X; the shaded area drops out. Top right: only the selected group — the narrower X range is immediately visible. Red line = regression in the selection, dashed blue = population line (same slope!). The spread bars below the plots show ±1 s_x.

Bottom curve: the Thorndike relationship r(u). For the current true correlation ρ, it shows how the observed correlation depends on the spread ratio u. At u = 1 (no restriction), r = ρ. To the left (u < 1, restriction), r drops; to the right (u > 1, expansion via extreme groups), r rises. The point marks your current state.

Selection mode: Top (classic selection via cutoff) · Extreme groups (top + bottom — increases r!) · Middle (middle range). Selection fraction: what % remains. ρ: true population correlation. Thorndike correction: back-calculates the corrected value from r_selected and u — it should hit ρ.

ρ (rho) — the true correlation in the overall population, i.e. the test's real validity.
r — the correlation observed in the selected group (what you actually measure).
sx — spread (standard deviation) of X in the overall population.
s′x — spread of X in the selected group (the prime ′ stands for "restricted").
u = s′x / sx — the spread ratio: how much of the original spread remains. u = 1 means no restriction, u < 1 restriction, u > 1 expansion (extreme groups).
U = 1/u = sx / s′x — simply the reciprocal of u; it only shows up because the correction formula computes "from restricted to overall" (≥ 1 under restriction).
b — the regression slope of Y on X; it stays unbiased under direct selection on X.
ρ (rho-hat) — the Thorndike-corrected estimate of ρ. The hat "^" always marks an estimated/computed value in statistics.

Attenuation: r = ρu / √(1 − ρ²(1 − u²))  ·  Correction: ρ = rU / √(1 + r²(U² − 1)) with U = 1/u = sx/s′x.

Thorndike, R. L. (1949). Personnel Selection: Test and Measurement Techniques. Wiley.
Sackett, P. R. & Yang, H. (2000). Correction for range restriction: An expanded typology. Journal of Applied Psychology, 85(1), 112–118.
Hunter, J. E., Schmidt, F. L. & Le, H. (2006). Implications of direct and indirect range restriction for meta-analysis methods and findings. Journal of Applied Psychology, 91(3), 594–612.