"p < 0.05" is one of the most repeated phrases in any intro statistics course, and also one of the most commonly misunderstood. Getting p-value interpretation right matters more than memorizing the cutoff — here's what it's actually telling you.
What a p-value actually measures
A p-value is the probability of seeing a result at least as extreme as what you observed, if the null hypothesis were true. It's a statement about how surprising your data would be under "nothing is going on" — not a statement about whether something is going on.
Why 0.05 is the cutoff (and why it's somewhat arbitrary)
A p-value below 0.05 means there's less than a 5% chance of seeing data this extreme if the null hypothesis were actually true — unlikely enough that most fields have agreed to treat it as evidence against the null hypothesis. But 0.05 isn't a law of nature; it's a widely-adopted convention, which is why some fields use stricter cutoffs.
What a small p-value does NOT mean
- It's not the probability the null hypothesis is true. This is the single most common misinterpretation in hypothesis testing — a p-value of 0.03 does not mean there's a 3% chance the null hypothesis is correct.
- It doesn't measure how big or important the effect is. A tiny, practically meaningless effect can still produce a very small p-value if the sample size is large enough.
- p = 0.049 is not meaningfully different from p = 0.051. Treating 0.05 as a hard line where one side is "real" and the other isn't misses the point of what the number represents.
Worked example
A study compares exam scores between students who used a new study app and those who didn't, and finds a p-value of 0.03 for the difference in average scores.
Interpretation: if the app truly had no effect, there would only be a 3% chance of seeing a score difference this large just from random variation. Since 0.03 < 0.05, this is typically reported as a statistically significant result.
What it doesn't tell you: how large the score improvement actually was, or whether it's large enough to matter practically — that requires looking at the effect size separately.
Common mistakes
Reporting significance without reporting effect size. A statistically significant result with a tiny practical effect can be misleading if effect size isn't mentioned alongside it.
Treating "not significant" as "no effect." A p-value above 0.05 means you don't have strong enough evidence to rule out the null hypothesis — not that you've proven there's no effect at all.
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