P-Value Interpretation

What is a p-value?

The p-value is the probability of observing results at least as extreme as the data, assuming $H_0$ is true.

• Small p-value ($p < \alpha$): The data is unlikely under $H_0$ → reject $H_0$.
• Large p-value ($p \geq \alpha$): The data is consistent with $H_0$ → fail to reject.

Step-by-step: $p = 0.03$, $\alpha = 0.05$

Step 1: $p = 0.03 < 0.05 = \alpha$.

Step 2: The result is statistically significant at the 5% level.

Step 3: We reject $H_0$.

Common misinterpretations

• ❌ "There's a 3% chance $H_0$ is true." (p-value is NOT the probability $H_0$ is true.)
• ❌ "There's a 97% chance the alternative is true." (Also wrong.)
• ✅ "If $H_0$ were true, there's a 3% chance of seeing data this extreme or more."

The α thresholds

• $p < 0.05$: "significant" (★)
• $p < 0.01$: "highly significant" (★★)
• $p < 0.001$: "very highly significant" (★★★)

These are conventions, not magic cutoffs. A p-value of 0.049 and 0.051 are practically the same.

Try it in the visualization

The test distribution is drawn. The p-value is shown as the shaded tail area. The α threshold is marked. When p < α, the shaded area falls inside the rejection region.