The null hypothesis: what it is and how to write one

The null hypothesis, written H₀, states that there is no effect, no difference or no relationship in the population. It is the position your test provisionally adopts and then attempts to overturn.

The alternative hypothesis, H₁, is what you are actually interested in: that there is a difference, an effect or a relationship. The two must be mutually exclusive and between them cover every possibility.

It seems perverse to start by assuming the opposite of what you believe. The reason is that only the null is specific enough to calculate with.

“There is no difference” specifies a single value: zero. That is enough to work out exactly how your test statistic would behave across repeated samples, and therefore how unusual your actual result is. “There is a difference”, by contrast, is compatible with infinitely many values — a difference of 0.1, or 5, or 400 — each producing a different distribution. There is nothing definite to compute against.

This asymmetry has a practical consequence for how a study is framed. If your research question is genuinely “are these two things equivalent?” — comparing a cheaper treatment against an established one, say — conventional significance testing is the wrong tool from the outset, and no sample size will fix it. Failing to reject the null cannot demonstrate equivalence however large the study grows. You need an equivalence design, specified as such before data collection, with a pre-agreed margin defining what counts as practically the same.

This asymmetry is why the conclusions are phrased unevenly. You may “reject the null”, but you may only ever “fail to reject” it. There is no test result that establishes the null is true.

One further point of precision. The null hypothesis is not a statement about your research being uninteresting, and it is not something you are trying to defend. It is a calculating device: a specific enough claim that the behaviour of your test statistic can be worked out exactly. This is why the null always contains an equality even when the alternative is directional. Writing H₀: μ₁ ≤ μ₂ against H₁: μ₁ > μ₂ looks asymmetric, but the calculation is performed at the boundary — at μ₁ = μ₂ — because that is the point in the null region where a false positive is most likely. If the test controls the error rate there, it controls it everywhere else in the null region too.

Symbols worth knowing

Notice that the null in every row above is a single, precise value or a boundary, while each alternative covers a range. That is the asymmetry described earlier, appearing in the notation. It is also why the alternative is never the thing being tested directly: there is no way to compute what your statistic would do under “the means differ by some unspecified amount”.

Writing them out in full

In a dissertation, state them in words as well as symbols:

H₀: There is no difference in mean exam performance between students taught by retrieval practice and students taught by spaced practice (μ₁ = μ₂).

H₁: There is a difference in mean exam performance between students taught by retrieval practice and students taught by spaced practice (μ₁ ≠ μ₂).

A one-tailed test roughly halves the p-value, which makes it tempting and makes switching to one after seeing the data a recognised form of p-hacking. It is only legitimate when a result in the opposite direction would be treated exactly as a null result would be — that is, when you genuinely would not care and would not report it.

Naming these “Type I” and “Type II” is unhelpful shorthand that obscures what they are, and it is worth translating them each time you use them. A Type I error means you announced a finding that is not there. A Type II error means you missed one that is. Both are failures of the study, but only the first tends to reach print, which is part of why the published literature over-represents effects that later shrink or vanish on replication.

Setting α = .05 means accepting a 1-in-20 false positive rate. Tightening it to .01 reduces that risk but makes the test harder to pass, so genuine effects are missed more often. The two error rates trade off directly against each other at any fixed sample size.

The trade-off is easiest to see at the extremes. Set α to zero and you will never make a false positive, because you will never reject anything — and you will therefore miss every real effect. Set it to one and you reject everything, catching every real effect and every spurious one alongside it. Every usable value sits between those two, and choosing one is a judgement about which mistake you would rather make.

Power is the escape route

Statistical power is 1 − β, the probability of detecting an effect that is genuinely there. The convention is 80%. Increasing the sample size is the only lever that reduces both error types at once — which is why power analysis belongs at the design stage rather than after the data are in.

This is the point examiners test most reliably, and it follows directly from the logic. The test computes how surprising your data would be if the null were true. A large p-value means the data are unsurprising under the null — but they may be equally unsurprising under a great many alternatives.

Suppose you find no significant difference with 20 participants per group. Your data are consistent with a true difference of zero. They are also consistent with a true difference of 3 points, or 6 points, which your study simply lacked the power to detect. Nothing in the result distinguishes those possibilities.

Everything in this guide depends on one condition: the hypothesis and the test were decided before the data were seen. A p-value calculated after the analysis was chosen to suit the result does not mean what it claims to mean.

Why the order matters so much

The p-value assumes a single, pre-specified test. Every additional decision made after seeing the data — a different outcome measure, an added covariate, a dropped outlier, a subgroup — multiplies the opportunities for a chance result to appear, while the reported p-value continues to assume there was only one. The stated error rate quietly becomes fiction.

It is worth separating two things that are routinely confused: the hypothesis and the research question. The research question is broad and may evolve — “does this intervention help?”. The hypothesis is the narrow, testable proposition committed to before the data arrive, tied to one outcome measured one way. A study can legitimately explore many questions while formally testing a single hypothesis, provided the write-up makes clear which is which.

What to write down before collecting data

This is the substance of a pre-registration, and it is increasingly expected in psychology, education and health research. Even where no formal registry is involved, a dated analysis plan in your supervisor's inbox serves the same function and is straightforward to produce.

1. Writing hypotheses about sample statistics

Use population parameters — μ, ρ, π — not x̄ or r. Hypotheses are claims about populations.

2. Saying you accept the null

You fail to reject it. The test can never establish that the null is true.

3. Choosing one-tailed after seeing the data

This halves the p-value on the strength of a result you have already seen. Decide the direction in advance and justify it.

4. Writing hypotheses after the analysis

Fitting the hypothesis to the finding invalidates the p-value, which assumes the test was specified beforehand.

5. Confusing the null with a claim you believe

The null is a working assumption to be tested, not a position you are defending.

6. Treating a non-significant result as proof of no effect

Check the confidence interval. A wide one means the study could not tell, which is a different finding entirely.