A p-Value Is a Probability About Samples, Computed in a World Where the Null Is True

You'll learnwhat a p-value is — a tail area of the null's own sampling distribution — how the alternative decides which tail counts, why the number moves when the sample does, and why every standard misreading is the definition with its assuming-clause dropped.

A company claims a 40% renewal rate and a manager suspects less. Of 250 customers, 88 renewed — a proportion of 0.352. Is that low enough to matter? The p-value answers a very specific version of that question: if the rate really were 0.40, how often would a random sample of 250 land at 0.352 or lower? The whole calculation happens inside the world that if builds — a curve centered at 0.40 with a spread computed from 0.40 — and the answer, about 6%, is an area of that curve's left tail. It is not the probability the claim is true, not the probability the result was chance, and not the probability of this exact sample; each of those misreadings is this animation's definition with one clause removed, and each gets computed here so you can watch it come out as a different number.

8 STEPS · 6 QUICK CHECKS · THE ASSUMING CLAUSE · TAIL AREAS · ONE- AND TWO-SIDED · v1

assuming p = 0.40 · z = −1.55 · left tail 0.0607 · both tails 0.1213

Before you start

What you're looking at

The distribution of sample proportions that a true rate of 0.40 would produce in samples of 250 — the null hypothesis's own world, drawn before any data arrive.

The question

The sample gave 0.352. How often would the null's world produce a result that far out — and what does that number mean, exactly?

Watch for

The word assuming. Every number on this page — the center, the spread, the shaded area — is computed inside the null's world, which is why none of them can be a probability that the null is true.

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