Sketch the Curve, Shade the Region, Then Compute

You'll learnwhy probability is area for a continuous variable, how a z-score turns any normal value into a position with a direction, where the 68-95-99.7 rule comes from and which distributions it does not describe, and how to run the calculation in both directions.

For a continuous variable, probability is area under a density curve, and the normal curve is the one whose areas are known the moment a value is converted to standard deviations from the mean. Scores with mean 500 and standard deviation 100 put a 650 at z = 1.5, and about 6.7% of scores sit beyond it. This animation computes every such area from the curve itself — including the 68, 95 and 99.7 the empirical rule recites — and then aims the rule at a distribution it does not describe: household incomes with mean 72,000 and standard deviation 55,000, where the rule claims 2.5% of households earn less than −38,000. Income stops at zero; the claim is off by every household in it. The habit the topic builds is mechanical and cheap: draw the curve, mark the value, shade the region asked for, and only then reach for the arithmetic.

8 STEPS · 6 QUICK CHECKS · AREA IS PROBABILITY · z-SCORES · 68-95-99.7 AND ITS LIMITS · v1

area is probability · z carries direction · 68-95-99.7 is the normal curve’s own

Before you start

What you're looking at

A normal curve for scores with mean 500 and standard deviation 100. Every probability in this topic is an area under this curve, and every area on screen is computed from it.

The question

How does a value become a z-score, a z-score become a shaded region, and a shaded region become a probability — and when does the normal model not apply at all?

Watch for

The income distribution in step 4: the empirical rule's interval reaches 38,000 below zero, and the computed areas show how far a skewed shape drifts from 68-95-99.7.

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