What is the null hypothesis?
Every p-value you have ever computed was calculated inside a world where the null hypothesis is true. Understanding what that assumption is doing is most of what it takes to read a result correctly.
7 min read · Last reviewed 7 August 2026
Null hypothesis definition
The null hypothesis, written H₀, is the claim that there is no effect,
no difference, and no relationship — that whatever pattern you can see in your data
is the work of ordinary random variation.
It is the position your test starts from and, unless the evidence is strong enough, the
position it stays in. The alternative hypothesis, H₁ or Hₐ, is
simply its negation: there is an effect.
Why testing works this way round
You cannot compute the probability of your data under "there is an effect," because that covers infinitely many possibilities — an effect of 0.1, of 5, of 400. There is no single distribution to compute against.
"There is no effect" is precise. It fixes the difference at exactly zero, which pins down a specific distribution the statistic must follow, which is what makes an exact tail probability computable at all. The whole apparatus exists because the null is the only hypothesis specific enough to calculate with.
So the p-value answers a narrow question: if the null were true, how often would data look at least this extreme? A small answer makes the null uncomfortable. It does not make the alternative probable — that would need a prior, which no p-value has. The p-value explainer works through why the two are different.
Stating it for each test
| Test | Null hypothesis |
|---|---|
| One-sample t | The population mean equals the target value |
| Two-sample t | The two population means are equal |
| Paired t | The mean difference is zero |
| Two proportions | The two population rates are equal |
| Correlation | The population correlation is zero |
| Chi-square independence | The two variables are independent |
| One-way ANOVA | All group means are equal |
Note the last one. ANOVA's null is that every mean is equal, so rejecting it tells you only that at least one differs somewhere — not which, and not how many. That is why a significant ANOVA is a beginning rather than a conclusion.
Note also that every null is a statement about the population, never about your sample. Your sample means will differ; the question is whether the populations do.
Reject, or fail to reject — never accept
A hypothesis test has exactly two possible outcomes, and neither is "the null is true."
- Reject H₀ — the data would be surprising if the null held, so the null is implausible.
- Fail to reject H₀ — the data are consistent with the null. They are also consistent with a real effect your study was too small to detect.
The clumsy phrase "fail to reject" is doing real work. A non-significant result cannot distinguish between no effect and an effect you lacked the power to see, so writing "there was no difference" claims something the test never established. The honest version reports the confidence interval: if it runs from −0.24 to 9.04, values up to 9 remain entirely compatible with your data, and calling that "no difference" is plainly wrong.
Absence of evidence is not evidence of absence. Proving equivalence requires a different design — an equivalence or non-inferiority test — not a failed significance test.
Does p less than 0.05 mean reject the null hypothesis?
Yes — that is the convention, and it is the whole decision rule. A p-value less than 0.05 means reject the null hypothesis at α = 0.05, because data at least this extreme would turn up less than 5% of the time if the null were true. Above 0.05, you fail to reject it.
The mirror-image phrasing, that a p-value less than 0.05 means you accept the null hypothesis, has the rule backwards twice over, and it is worth separating the two errors:
- The direction is inverted. A small p-value is evidence against the null, not for it. Below your α you reject; it is a p-value above α that leaves the null standing.
- "Accept" is not an available outcome. Even when p is large, the result is "fail to reject" — the data are consistent with no effect, and equally consistent with a real effect too small for your sample to detect.
So there is no combination of p-value and α that lets you accept H₀. You reject it or you do not, and neither verdict is a proof of it. What "p < 0.05" does and does not license you to claim is worked through in detail in what a p-value less than 0.05 actually means, and the threshold itself is a choice rather than a law — see choosing a significance level.
Testing a null hypothesis with a p-value calculator
Turning a stated null into a decision takes three steps, and a p-value calculator handles the middle one.
- State H₀ and pick α before you look at the data. Both choices become meaningless if you make them after seeing the result.
- Compute the p-value. If your software has already given you a test statistic, the p-value calculator converts it — with the tail area shaded, which is a literal picture of the probability the null assigns to data like yours. If you have raw numbers instead, the t-test calculator and the A/B test calculator go straight from data to p.
- Compare and report. Below α you reject H₀; above it you fail to reject. Either way, report the exact p-value with an effect size and a confidence interval, since the p-value alone never says how large the effect is.
The equivalent route is to compare your statistic against a threshold instead of computing a tail area — the critical value calculator gives that boundary. The two always reach the same verdict.
Directional nulls
A two-tailed test has the null "the means are equal" and treats a difference in either direction as evidence against it. A one-tailed test narrows the alternative to a single direction, so the null becomes "no effect, or an effect the other way."
The direction has to be committed to before seeing the data. Choosing it afterwards halves your p-value without adding any evidence, and roughly doubles your false positive rate. The one-tailed vs two-tailed guide sets out the cases where it is legitimate — they are narrower than most people assume.
What the null is not
It is not what you believe. Researchers usually expect an effect; the null is a device for testing that expectation, not a statement of anyone's opinion.
It is not "nothing happened." It is the specific claim that a particular population parameter takes a particular value — usually zero, but not always. Testing whether a coin is fair makes the null p = 0.5.
A small p-value does not make it false. It makes it improbable that data this extreme would arise if it were true. Those are different statements, and the gap between them is where most p-value misreading lives.