Mistake Master
Probability Is Area, and a z-Score Only Tells You Where to Shade
You'll learnwhy probability is area for a continuous variable, how a z-score turns a value into a position with a direction, where 68-95-99.7 comes from and which shapes it does not describe, and how to run the calculation backward from an area to a value.
Scores have mean 500 and standard deviation 100, and the whole topic is one habit: draw the curve, shade the region you were asked about, and only then compute. This animation starts with a strip of the curve and narrows it until its area vanishes, which is why a single value has probability zero. Then a 650 becomes z = 1.5, and the answer turns out to be the shaded tail past that position rather than the 1.5 itself. The 68, 95 and 99.7 everyone recites are measured off the curve here, not looked up — and then aimed at household incomes with mean 72,000 and standard deviation 55,000, where the rule builds an interval starting 38,000 below zero and the true areas are nowhere near what it promised. It finishes on the two calculations that run the other way: an area back to a value, and two different tests onto one ruler. Twelve short steps; press Next for each one.