Norms & standardized scores

A raw score of 42 tells you nothing. Forty-two out of what? Compared to whom? Standardized scores solve this by locating a person on the normal curve relative to a norm group — and the exam demands you convert between them fluently, because the whole family is just the same bell curve wearing different clothes.

Start with the normal curve and its landmarks: it's symmetric, and the 68-95-99.7 rule says about 68% of scores fall within ±1 SD of the mean, 95% within ±2 SD, and 99.7% within ±3 SD. Every standardized score is anchored to this curve.

The z-score is the mother score: mean 0, SD 1. A z of +1 means one SD above the mean. Because z-scores carry decimals and negatives, we rescale them. The T-score has mean 50, SD 10, computed as T = 50 + 10z — so z=0 is T=50, z=+1 is T=60 (used on the MMPI and many clinical inventories). Stanines ('standard nines') compress scores into 1-9 with a mean of 5 and SD of 2. Percentile ranks tell you the percent of the norm group scoring at or below a person — the 50th percentile is the median, and it lines up with z=0 and T=50. The deviation IQ uses mean 100, SD 15.

The anchor points to burn in: z = 0 = T-score 50 = 50th percentile = stanine 5 = IQ 100 — all the dead center of the same curve. One caution: percentiles are not equal-interval, so they bunch up in the middle and stretch out at the tails.

Clinical Pearl: Memorize the center-line conversion — z=0, T=50, 50th percentile, stanine 5, IQ 100 all mark the exact same point — and remember T = 50 + 10z lets you jump from any z straight to a T-score.