Two cohorts, two z-scores, and one rule that only works on a bell

Maths Standard 2 · z = (x − μ) / σ · a z-score is a POSITION, a percentage is an AREA

Shape

Shade

Subject A

axis: mark, then z · height fitted

Mean, μ 60 Standard deviation, σ 12

Mark, x 72

Subject B

axis: mark, then z · height fitted

Mean, μ 75 Standard deviation, σ 5

Mark, x 79

Does the 68/95/99.7 rule hold for this shape?

these percentages depend only on the SHAPE, so they are the same for A and B

Region	The rule claims	This data actually has	Verdict

Predict before you look

Kim scored 74 in Subject A, where the mean is 65 and the standard deviation is 6, and 81 in Subject B, where the mean is 70 and the standard deviation is 11. Which was the better performance relative to the cohort ?

Table check: is this build's normal CDF the same one as the printed z-table?

z	this build	standard normal table

Method: Abramowitz & Stegun 26.2.17 (the Zelen and Severo rational approximation), stated absolute error below 7.5 × 10⁻⁸ for every z. The largest error over the four rows above is under 1 × 10⁻⁷, which is far below the 4-decimal-place resolution of a printed table. Every percentage on this page comes from that one function, so if these four rows agree with your table, the shaded areas do too.

Ready. Move a mark, shade a region, then switch the shape to skewed and watch the 68/95/99.7 row fail.