Mistake Master
Center, Spread, Shape — and What Each One Costs
You'll learnthe three facts that make the sampling distribution of a sample proportion usable — centered at p, spread by the square root formula, approximately normal when the counts are large — which condition buys each fact, how to turn a question about p-hat into a z-score, and how to catch the standard error rebuilt from the wrong parts.
The sample proportion has a distribution of its own, and this page draws it exactly — all 101 possible values of p-hat for a sample of 100, each with its precise probability — so every formula the topic teaches can be checked against the thing it summarises. The spread formula passes: the square root of p(1−p)/n says 0.0490 and the enumeration agrees to six decimals, while the three popular wrong builds miss by factors of ten and twenty in both directions. The normal shape is a different kind of claim: an approximation with a measurable error, about 0.02 at p = 0.60 with n = 100 and eight times smaller when the sample quadruples. And at p = 0.02, where a sample of 100 expects only two successes, the normal picture simply breaks — it promises a tail three times too small and parks 7.7% of its area below zero, where no sample can land. One condition per fact, and the page shows what each one is protecting you from.