Score Converter

All score types are first converted to a z-score (the universal intermediate), then from the z-score to every other type using standard linear transformations:

• T-Score: T = (z × 10) + 50
• Standard Score: SS = (z × 15) + 100
• Scaled Score: Scaled = (z × 3) + 10

Z-Score → Percentile (normalCDF)

Converting a z-score to a percentile requires the cumulative distribution function (CDF) of the standard normal distribution—i.e., the probability that a normally distributed value falls at or below z.

We use the Abramowitz & Stegun polynomial approximation (formula 26.2.17), which achieves an absolute error less than 7.5 × 10⁻⁸. It computes:

k = 1 / (1 + 0.2316419 × |z|)
Φ(z) ≈ 1 − φ(z) × k(b₁ + k(b₂ + k(b₃ + k(b₄ + k·b₅))))

where φ(z) is the standard normal PDF and b₁…b₅ are fixed coefficients. For negative z, symmetry gives Φ(z) = 1 − Φ(|z|).

Reference: Abramowitz & Stegun, Handbook of Mathematical Functions, p. 932.

Percentile → Z-Score (inverseCDF)

The reverse—converting a percentile to a z-score—requires the inverse (quantile function) of the normal CDF. We use a rational approximation with two regions:

• Central region (|p − 0.5| < 0.42): a rational polynomial in (p − 0.5)² using coefficients a₁…a₄ and b₁…b₄.
• Tail region: a polynomial in r = ln(−ln(p)) using nine coefficients c₁…c₉, with sign correction for the lower tail.

Reference: Beasley & Springer, rational approximation for the inverse normal CDF (pp. 16).