P-values, explained with examples

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

A p-value measures how surprising your data would be if the null hypothesis were true. It is the number that decides most significance tests, and also one of the most misunderstood ideas in statistics.

The definition

A p-value is the probability, assuming the null hypothesis H₀ is true, of getting a sample result at least as extreme as the one actually observed, in the direction of the alternative hypothesis.

Every part of that sentence matters. “Assuming H₀ is true”: the calculation imagines a world where nothing unusual is going on. “At least as extreme”: we count the observed result and anything further from what H₀ predicts. “In the direction of H_a”: a one-sided test looks in one tail; a two-sided test looks in both.

The logic of a significance test

Suppose a coin is claimed to be fair, and you get 64 heads in 100 flips. If the coin really were fair, how often would you see 64 or more heads? If that would happen only rarely, you have evidence that the coin is not fair. A small p-value says: “this result would be unusual if H₀ were true”. So either something very unlikely happened, or H₀ is wrong. A significance test takes the second option when the p-value is small enough.

Comparing with the significance level

Before collecting data, choose a significance level α, usually 0.05. Then:

• If p-value < α, reject H₀. There is convincing evidence for H_a.
• If p-value ≥ α, fail to reject H₀. There is not convincing evidence for H_a.

Never say “accept H₀”. A large p-value means the data are consistent with H₀, not that H₀ has been proved true. The sample may simply have been too small to detect a real effect.

Worked examples

Example 1: a one-sided test for a proportion

A school claims 50% of students walk to school. A researcher suspects it is more. In a random sample of 100 students, 58 walk. Find the p-value.

• H₀: p = 0.5. H_a: p > 0.5.
• Standard deviation of p̂ under H₀: √(0.5 × 0.5/100) = 0.05. z = (0.58 − 0.5)/0.05 = 1.6.
• p-value = P(Z ≥ 1.6) ≈ 0.0548.

Example 2: the decision

Using α = 0.05, what should the researcher conclude?

• 0.0548 > 0.05, so fail to reject H₀.
• There is not convincing evidence that more than half of the school's students walk. The result is close to the cut-off, so a larger sample would help.

Example 3: a two-sided test

If the researcher had tested H_a: p ≠ 0.5 with the same data, what would the p-value be?

• Two-sided: count both tails, 58% or more and 42% or fewer.
• p-value ≈ 2 × 0.0548 = 0.1096.

Example 4: interpreting a p-value in context

A test of whether a new diet lowers cholesterol gives a p-value of 0.012. Interpret it.

• Assuming the diet has no effect on mean cholesterol, there is a 0.012 probability of getting a sample mean reduction at least as large as the one observed, by random chance alone.
• This is the interpretation AP readers want: it states the assumption, the probability, and the “at least as extreme” idea, all in context.

Example 5: a p-value from a t-test

A random sample of 25 bags of crisps has mean mass 31.0 g and standard deviation 2.0 g. Test whether the mean is more than the stated 30 g.

• H₀: μ = 30. H_a: μ > 30. The population standard deviation is unknown, so use a t-test with 24 degrees of freedom.
• t = (31.0 − 30)/(2.0/√25) = 1.0/0.4 = 2.5.
• p-value = P(t₂₄ ≥ 2.5) ≈ 0.0098. At α = 0.05, reject H₀: there is convincing evidence that the mean mass of all bags is more than 30 g.

What a p-value is not

• It is not the probability that H₀ is true.
• It is not the probability that the result happened by chance.
• It does not measure how big or important an effect is. A tiny, unimportant difference can have a very small p-value if the sample is huge.
• A p-value above 0.05 does not prove there is no effect.

Statistical significance versus practical importance

With a large enough sample, even a trivial difference becomes statistically significant. A medicine might lower blood pressure by 0.5 points with p = 0.001: real, but not useful. That is why a confidence interval, which shows the size of the effect, is a valuable companion to a p-value. Always ask both questions: is there evidence of an effect, and is the effect large enough to matter?

Why 0.05?

The 5% level is a convention, not a law of nature. It balances the risk of a false alarm against the risk of missing a real effect. In medicine or engineering, where a false alarm could be costly, researchers often use 0.01. In early exploratory work, 0.10 is sometimes used. The key rule is to choose α before seeing the data.

Where p-values come from in practice

On the AP exam you find p-values with a calculator, a table, or computer output. Every test follows the same pattern: state the hypotheses, check conditions, calculate a test statistic (z, t or χ²), find the p-value from the matching distribution, compare with α, and conclude in context.

Common mistakes

• Interpreting the p-value as the probability that H₀ is true.
• Writing “accept H₀” instead of “fail to reject H₀”.
• Forgetting to double the one-tail area for a two-sided test.
• Giving a conclusion without context or without linking the p-value to α.
• Choosing α after seeing the p-value.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Are p-values on the AP Statistics exam?

Yes. They appear in every significance test in Units 6 to 9, and interpreting a p-value in context is a frequent free-response task.

Are p-values in A level Maths?

A level Maths hypothesis tests often compare a probability with the significance level, which is a p-value approach, though some boards use critical regions instead.

Key takeaways

• A p-value assumes H₀ is true.
• It is the probability of a result at least as extreme as the one observed.
• Small p-values give evidence against H₀.
• Compare with α chosen in advance; conclude in context.
• Significance is not the same as practical importance.

Related guides

• AP Statistics hypothesis testing
• Type I and Type II errors
• Central limit theorem