The Normal Distribution
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalA continuous variable's probability is area under its density curve, whose total area is 1, so a single value carries probability 0 and only intervals have probability. A normal distribution is fixed by $\mu$ and $\sigma$, and $z = \frac{x - \mu}{\sigma}$ converts any value to a position in standard deviations from the mean, with the sign carrying direction. The empirical rule then gives approximately 68%, 95%, and 99.7% of values within one, two, and three standard deviations. Working backward, $x = \mu + z\sigma$ turns a percentile into a value: the 90th percentile of a $N(500, 100)$ distribution is $500 + 1.28(100) = 628$.
Two errors define this topic. The model is applied where it does not belong: the empirical rule laid over a right-skewed income distribution predicts 95% of households between $-38{,}000$ and 182,000, and a negative income is the proof. And the mechanics slip for lack of a sketch: a z-score reported as a probability, a cumulative area handed back when the question asked for the upper tail, a negative sign dropped so the answer lands in the wrong tail. Drawing the curve, marking the value, and shading the asked-for region before computing catches all of them.
The work
3 ways in · any order
Lesson
The Normal Distribution
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Treats probability as area under a density curve, standardizes with z-scores that keep their sign, fixes the empirical rule to the normal model it belongs to, and builds the sketch-shade-compute habit for both directions of the calculation.
Diagnostic
10-item topic check
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Ten items on the normal model: the empirical rule applied to skewed data, z-scores read as probabilities, cumulative areas returned where an upper tail was asked for, and signs dropped. Take it cold to find your habit, or after the lesson to check it is gone.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.