What a z-score cuts off
The shaded area is the percentile: the share of people scoring at or below IQ 130 on an SD-15 scale.
One z, three IQs: the SD decides the number you're told
A "148" on a Cattell-scale report and a "130" on the WAIS describe the same rank (z = +2). Quote the z, or the scale with the number, and claims stop talking past each other.
Standardising a raw score against its norm group
Test publishers give a few thousand people the test, then record the mean and SD of raw scores for each age band. Your raw score is placed against that band. Real manuals use smoothed lookup tables rather than one formula, but the idea is this.
z = (42 − 34) / 6+1.33IQ = 100 + 15 × 1.33120percentile = Φ(1.33)90.9%Two z-scores of +2 don't average to a composite of +2
Scoring high on two tasks that only partly overlap is rarer than scoring high on either. So the composite of two subtests lands further out than their average. Tellegen and Briggs (1967) gave the correction for building a composite from correlated scores: zc = (z₁ + z₂) / √(2 + 2r).
That's why a full-scale IQ is often more extreme than the index scores it is built from, and why averaging subtests by hand runs too close to 100.
- Tellegen, A., & Briggs, P. F. (1967). Old wine in new skins: Grouping Wechsler subtests into new scales. Journal of Consulting Psychology, 31(5), 499–506.
- Wechsler, D. (2008). WAIS-IV Technical and Interpretive Manual. Pearson. Composite scores: mean 100, SD 15.
- Any introductory statistics text on standard scores and the normal distribution (e.g., the 68–95–99.7 rule).