Null Hypothesis Explained

What Is the Null Hypothesis?

The null hypothesis H₀ is a specific, testable claim about a population parameter. It represents the "status quo" — the assumption that nothing interesting is happening. Statistical tests are designed to measure how much evidence the data provides against H₀.

The word "null" comes from the Latin "nullus" meaning "none" — as in "null effect," "null difference," or "null relationship." H₀ almost always takes the form of equality: μ = 0, μ₁ = μ₂, ρ = 0, or observed frequencies = expected frequencies.

Key property: H₀ must be specific enough to generate a probability distribution. "There is no difference between groups" is testable. "Something might be happening" is not. This specificity is what makes statistical testing possible.

H₀ vs H₁ — The Alternative Hypothesis

The alternative hypothesis H₁ (also written Hₐ) is what you believe may be true if H₀ is false. It is the claim you are trying to find support for.

H₁ can be directional (one-tailed) or non-directional (two-tailed). Example: testing whether a new drug lowers blood pressure. Two-tailed: H₀: μ_new = μ_control; H₁: μ_new ≠ μ_control (any difference). One-tailed: H₀: μ_new ≥ μ_control; H₁: μ_new < μ_control (specifically lowers BP).

Important: you never directly "prove" H₁. You can only find evidence against H₀. A statistically significant result means "the data are inconsistent with H₀" — it does not prove H₁ is true.

How to Write a Null Hypothesis

Step 1: Identify the population parameter. Is it a mean (μ), proportion (p), correlation coefficient (ρ), or variance (σ²)?

Step 2: State the "no effect" version. H₀ almost always uses = (equality). Examples: μ = 50, μ₁ − μ₂ = 0, ρ = 0, p = 0.5.

Step 3: Write H₁ as the logical complement. Two-tailed: μ ≠ 50. One-tailed: μ > 50 or μ < 50. Choose one-tailed only if you have a strong directional prediction made before seeing the data.

Step 4: Check specificity. H₀ must specify a single value (for means and proportions) or a general structure (for chi-square and ANOVA). "The drug has no effect" alone is not testable — "the drug does not change mean BP (μ_treated − μ_control = 0)" is.

Rejecting vs Failing to Reject — What It Means

"Rejecting H₀" means the data are statistically inconsistent with H₀ at the chosen significance level α. If p ≤ α, you have sufficient evidence to reject H₀ in favor of H₁.

"Failing to reject H₀" does NOT mean H₀ is true. It means the data do not provide sufficient evidence against H₀. This distinction is crucial: absence of evidence is not evidence of absence.

Analogy: a court verdict. "Not guilty" does not prove the defendant is innocent — it means the evidence was insufficient for conviction. Similarly, "fail to reject H₀" means "insufficient statistical evidence," not "H₀ is confirmed."

You should never say "we accept H₀" or "we proved H₀." The correct language is "we fail to reject H₀" or "there is insufficient evidence to reject H₀."

Common Mistakes When Stating Hypotheses

Mistake 1: Vague hypotheses. "Students will perform better with method A" is a prediction, not a testable hypothesis. Correct: "Mean exam score with method A (μ_A) is greater than with method B (μ_B): H₀: μ_A = μ_B, H₁: μ_A > μ_B."

Mistake 2: Choosing one-tailed to get a smaller p-value. One-tailed tests should only be used when you have a genuine directional prediction before collecting data. Switching to one-tailed after seeing results (because p = 0.06 two-tailed and you want p < 0.05) is p-hacking.

Mistake 3: Confusing H₁ with H₀. Researchers often care deeply about H₁ (the effect they hope to find), leading them to accidentally "test" their preferred hypothesis instead of the null.

Mistake 4: Stating H₀ about the sample, not the population. "The sample mean is 50" is an observation, not a hypothesis. H₀ must be about the population parameter: "The population mean μ = 50."