Z-Score (Standard Score)

A z-score strips the units off a measurement and reports it in standard deviations: subtract the mean, divide by the standard deviation, and a 130 IQ (μ = 100, σ = 15) becomes z = (130 − 100)/15 = 2.0, exactly the same standing as a 700 on an SAT section scaled to μ = 500, σ = 100. That common currency is the whole point: it lets you compare a bench-press result to a reading score. The Belgian astronomer Adolphe Quetelet was the first to push this idea hard: in the 1830s he applied the astronomers' error curve to chest circumferences of Scottish soldiers, inventing l'homme moyen and, with it, the habit of measuring people against a mean and a spread.

The trap is forgetting that z-scores only carry probability meaning when the distribution is roughly normal. A z of 2.0 is about the 98th percentile in a bell curve, but in a badly skewed distribution (household income, say) the same z can be nowhere near the top 2%. Run it backwards to recover a raw score: on a test with μ = 72 and σ = 8, the cutoff for the top decile (z ≈ 1.28) is x = 72 + 1.28 × 8 ≈ 82.2.