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What is a p-value?

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University Course Reader · STEM

The treatment group differed significantly from control (p-value = 0.03), though the effect size was modest.

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Overview

When a study finds a difference, there are two possible causes: something real, or luck. The p-value measures how easily luck alone could produce a difference that big, so a small p-value means luck would rarely manage it. It does not tell you the chance the finding is true.
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Overview

A p-value isn't the chance you're right. It's the chance that pure luck could fake a result this impressive when the thing you tested does nothing. A small number means luck would struggle to pull it off. 😎

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Detail

Take a sleep-drug trial where the treated group fell asleep 12 minutes faster than the sugar-pill group. The boring assumption, called the null hypothesis, is that the drug does nothing and the 12 minutes is ordinary luck, since two groups of people never come out exactly equal. The p-value answers one question: if the drug truly did nothing, how often would luck alone open a gap of 12 minutes or more? If luck manages that only 3 times in 100 repeats of the study, the p-value is 0.03, and something real looks likely. Most fields call anything under 0.05 significant, but that cutoff is a convention, and clearing it does not mean the effect is big enough to matter. One caution: test 20 useless drugs at that cutoff and about one will pass by luck alone, so a single lucky pass proves little.
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Detail

Some sleep drug supposedly works: the group that took it dozed off 12 minutes faster than the sugar-pill group. Boring explanation first: the drug does nothing, and 12 minutes is just the random gap you'd get between any two groups of people. The p-value asks how often that random gap alone would reach 12 minutes, and if the answer is 3 in 100 reruns of the study, your p-value is 0.03. The sacred 0.05 cutoff is just a number a statistician picked a century ago, and clearing it doesn't mean the effect is big enough to care about. Watch for the classic scam: test twenty useless drugs and roughly one will sneak under 0.05 by dumb luck, which is why one lucky pass proves nothing. 😎

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Analogy

A friend rolls three sixes in a row and calls it luck. A fair die does that about once in 216 tries, and that once-in-216 chance is the p-value of his claim. It is small, so the die being fair stops being the easy explanation, though rare things do happen.
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Analogy

Your buddy swears he can taste the difference between the two colas and nails 9 cups out of 10. Pure guessing would hit 9 or more only about once in a hundred tries, and that once-in-a-hundred is his p-value. It's small enough that lucky guessing is no longer the believable story.
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Formal definition — The same term, explained the usual way

The p-value is the probability, computed under the null hypothesis, of obtaining a test statistic at least as extreme as the observed value. It quantifies incompatibility between data and the null model, and does not represent the probability that the null hypothesis is true, nor does it measure effect magnitude. Multiple comparisons inflate the family-wise error rate unless corrected.

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