A null hypothesis (H0) is the starting assumption a statistical test tries to reject: that there is no difference between groups, no relationship between variables or no effect of a treatment. The alternative hypothesis (H1 or Ha) states what the researcher expects instead. For example, H0: mean exam scores are the same for students who study with flashcards and students who don't. H1: the mean scores differ.
The 30 pairs below come from eight fields, each with the test that usually fits; the right test for your study still depends on its design and data.
| Field | Pairs |
|---|---|
| FieldMedicine and health | Pairs1–5 |
| FieldPsychology | Pairs6–9 |
| FieldEducation | Pairs10–13 |
| FieldBiology and ecology | Pairs14–17 |
| FieldBusiness and marketing | Pairs18–21 |
| FieldEconomics and social science | Pairs22–24 |
| FieldEngineering and manufacturing | Pairs25–27 |
| FieldTechnology and computing | Pairs28–30 |
Null vs alternative hypothesis
The two always come as a pair. The NIST/SEMATECH e-Handbook of Statistical Methods, published by the US National Institute of Standards and Technology, says a statistical test requires both: the null, the conjecture being tested, and the alternative, what you conclude if the data rule the null out.
| Null hypothesis (H0) | Alternative hypothesis (H1 or Ha) | |
|---|---|---|
| What it says | Null hypothesis (H0)No effect, no difference, no relationship, or one specific value | Alternative hypothesis (H1 or Ha)There is an effect, a difference or a relationship |
| Symbols | Null hypothesis (H0)=, or ≤ or ≥ in some one-tailed setups | Alternative hypothesis (H1 or Ha)≠ for two-tailed tests, < or > for one-tailed |
| Role in the test | Null hypothesis (H0)Assumed true while the test is calculated | Alternative hypothesis (H1 or Ha)Concluded only if the data are unlikely under the null |
| Possible outcomes | Null hypothesis (H0)Rejected, or not rejected | Alternative hypothesis (H1 or Ha)Supported when the null is rejected |
| Typical wording | Null hypothesis (H0)“…is the same as…”, “…is not related to…” | Alternative hypothesis (H1 or Ha)“…differs from…”, “…is greater than…”, “…is related to…” |
The two possible decisions are easiest to see in a sentence. Here is a result that didn't reach significance, written up two ways:
Overclaims: “The results showed that tutoring has no effect on pass rates.”
Accurate: “We failed to reject the null hypothesis of equal pass rates: 62% of students who attended tutoring passed, against 58% of those who did not (p = 0.19). The 95% confidence interval for the difference, −2 to 10 percentage points, is consistent with anything from a slightly lower pass rate to one 10 points higher among students who attended.”
The accurate version names the decision, gives the size of the difference and shows how wide the uncertainty is. Failing to reject means the data didn't rule the null out. It is not evidence that the two groups pass at the same rate, a difference explained in why you never accept the null, below.
The direction of the alternative sets the tails of the test. Penn State's online statistics review uses a mean grade point average of 3 to show all three cases. It keeps the equals sign in the null each time; NIST's handbook writes a one-sided null as an inequality instead, with a mean bulb lifetime “greater than or equal to 500 hours.” Both conventions put the equality in the null.
| Test | H0 | H1 | Use it when |
|---|---|---|---|
| TestRight-tailed | H0μ = 3 | H1μ > 3 | Use it whenOnly an increase matters |
| TestLeft-tailed | H0μ = 3 | H1μ < 3 | Use it whenOnly a decrease matters |
| TestTwo-tailed | H0μ = 3 | H1μ ≠ 3 | Use it whenA change in either direction matters |
Examples from medicine and health
Treatments and exposures compared on a clinical measure.
- H0: The mean drop in systolic blood pressure after 12 weeks is the same for patients on the new drug and patients on placebo (μ₁ = μ₂). H1: The mean drop differs (μ₁ ≠ μ₂). Two-sample t-test, two-tailed.
- H0: The share of patients who develop a surgical-site infection is the same with and without an antiseptic wash before surgery (p₁ = p₂). H1: The shares differ (p₁ ≠ p₂). Chi-square test or two-proportion z-test.
- H0: Daily step count is not correlated with resting heart rate in adults over 40 (ρ = 0). H1: The two are correlated (ρ ≠ 0). Pearson correlation test.
- H0: The mean length of stay is the same in three hospital wards (μ₁ = μ₂ = μ₃). H1: At least one ward's mean differs. One-way ANOVA. The alternative is “at least one,” not “all three differ.”
- H0: Mean fasting blood glucose is the same before and after an eight-week walking program for the same participants (μd = 0, where d is the after-minus-before difference). H1: It changes (μd ≠ 0). Paired t-test.
Examples from psychology
Conditions in an experiment, or behavior across groups.
- H0: Mean recall is the same for participants who study in silence and participants who study with music (μ₁ = μ₂). H1: Mean recall differs (μ₁ ≠ μ₂). Two-sample t-test.
- H0: Mean anxiety scores are the same after four weeks of a mindfulness app, a journaling task or a waitlist (μ₁ = μ₂ = μ₃). H1: At least one group's mean differs. One-way ANOVA.
- H0: Hours of sleep have no linear relationship with next-day reaction time (β₁ = 0). H1: They do (β₁ ≠ 0). Simple linear regression.
- H0: Whether a bystander helps is independent of how many other people are present, across three group sizes. H1: Helping depends on group size. Chi-square test of independence.
Examples from education
Teaching methods and programs, measured in scores and pass rates.
- H0: Mean final exam scores are the same in sections that give weekly quizzes and sections that don't (μ₁ = μ₂). H1: The means differ (μ₁ ≠ μ₂). Two-sample t-test; if whole sections are assigned, use a model that accounts for students being grouped in sections.
- H0: Pass rates in a gateway math course are the same for students who attend tutoring and students who don't (p₁ = p₂). H1: The pass rates differ. Chi-square test. Students choose tutoring, so rejecting H0 shows a difference, not that tutoring caused it.
- H0: Mean reading fluency is the same at the start and end of a summer reading program (μd = 0). H1: It differs (μd ≠ 0). Paired t-test.
- H0: Students' choice of major is independent of whether either parent attended college. H1: Choice of major depends on parents' education. Chi-square test of independence.
Examples from biology and ecology
Growth, inheritance and habitats. Number 15 is the case where the null is a theory's prediction, not “no effect.”
- H0: Mean plant height after four weeks is the same with and without fertilizer (μ₁ = μ₂). H1: Mean height differs (μ₁ ≠ μ₂). Two-sample t-test.
- H0: In offspring of two parents that each carry one dominant and one recessive allele, for a trait with complete dominance, the dominant and recessive forms appear in a 3:1 ratio. H1: The offspring don't fit a 3:1 ratio. Chi-square goodness-of-fit test. Here the researcher usually hopes not to reject H0.
- H0: Urban and rural habitats have the same mix of bird species. H1: The mix differs. Chi-square test of independence on counts from survey plots.
- H0: Stream temperature is not correlated with dissolved oxygen (ρ = 0). H1: The two are correlated (ρ ≠ 0). Pearson correlation test.
Examples from business and marketing
A/B tests, claims and spending, where the data are often large.
- H0: Landing pages A and B have the same conversion rate (pA = pB). H1: The rates differ (pA ≠ pB). Two-proportion z-test.
- H0: Mean delivery time is at most 30 minutes, as the company advertises (μ ≤ 30). H1: Mean delivery time is longer than 30 minutes (μ > 30). One-sample t-test, right-tailed: a consumer group testing the claim cares only if deliveries are slower.
- H0: Mean customer satisfaction is the same for three store layouts (μ₁ = μ₂ = μ₃). H1: At least one layout's mean differs. One-way ANOVA, or Kruskal–Wallis for ratings on a short scale.
- H0: Monthly ad spending has no linear relationship with monthly sales (β₁ = 0). H1: It does (β₁ ≠ 0). Simple linear regression.
Examples from economics and social science
Surveys and before-and-after records. Number 23 tests a single value.
- H0: Support for a new transit tax is independent of age group, across four age groups. H1: Support depends on age group. Chi-square test of independence.
- H0: The mean rent for one-bedroom apartments in the city equals the $1,450 reported in last year's housing study (μ = 1,450). H1: The mean rent differs (μ ≠ 1,450). One-sample t-test, two-tailed.
- H0: For the same commuters, mean commute time is the same before and after a rapid bus line opens (μd = 0). H1: It changes (μd ≠ 0). Paired t-test.
Examples from engineering and manufacturing
Quality control, where the null is often a target value. Numbers 25 and 26 are the two examples in NIST's handbook.
- H0: Photomasks from a production process have a mean linewidth of 500 micrometers (μ = 500). H1: The mean linewidth is not 500 micrometers (μ ≠ 500). One-sample t-test, two-tailed.
- H0: The mean lifetime of a production lot of light bulbs is at least 500 hours (μ ≥ 500). H1: The mean lifetime is less than 500 hours (μ < 500). One-sample t-test, left-tailed.
- H0: Two filling machines have the same variance in fill weight (σ₁² = σ₂²). H1: The variances differ. F-test for equal variances, or Levene's test if the weights aren't normally distributed.
Examples from technology and computing
Software performance and quality, often with skewed or paired data.
- H0: Page load times on the old and new servers come from the same distribution. H1: Load times on one server tend to be longer. Mann–Whitney U test, a common choice because load times are often skewed.
- H0: Bugs per 1,000 changed lines occur at the same rate under the old and new code-review policies. H1: The rates differ. A test comparing two Poisson rates, or Poisson regression.
- H0: Two spam filters have the same misclassification rate on incoming mail. H1: Their rates differ. McNemar's test on one set of 5,000 test messages, because both filters judge the same messages.
How to write a null hypothesis
Work from the research question to the symbols, and write everything down before you collect the data.
- Start from the research question
“Does tutoring change pass rates in the gateway math course?” The question names the groups (with and without tutoring) and the outcome (passing). The hypothesis examples show how to get from a question to a research hypothesis.
- Name the population and the parameter
Hypotheses are claims about a population, so they use population values: a mean (μ), a proportion (p), a correlation (ρ). Here it is the pass rate among all students who take the course, not the rate in your sample.
- Write the alternative first
The alternative is usually your research hypothesis, the effect you expect: pass rates differ between the two groups.
- Write the null as its opposite
No difference, no relationship, no effect, with the equality: the pass rates are the same (p₁ = p₂).
- Choose the tails before you see the data
Use a one-tailed test only if a result in the other direction would be of no consequence. UCLA's statistics consulting group puts the rule plainly: switching to a one-tailed test after a two-tailed test failed to reject the null is not appropriate, and neither is choosing one just to reach significance.
- Set alpha and name the test
Choose the significance level (α) now. NIST's handbook notes that values of 0.1, 0.05 and 0.01 are common; 0.05 means the test will reject a true null 5% of the time. Then name the test that fits your data, as in the examples above.
Null and alternative, in words and symbols
H0: In [population], there is no difference in [outcome]
between [group A] and [group B]. (μA = μB)
H1: In [population], [outcome] differs
between [group A] and [group B]. (μA ≠ μB)
Test: [test name], two-tailed, α = 0.05
For a relationship:
H0: In [population], [X] is not related to [Y]. (ρ = 0)
H1: In [population], [X] is related to [Y]. (ρ ≠ 0)What a p-value tells you
The NIST/SEMATECH handbook defines the p-value as the probability of a test statistic at least as extreme as the one observed, given that the null hypothesis is true. A small p-value means your data would be surprising in a world where the null holds, which counts as evidence against it, so you compare p with the α you set in advance and reject the null when p is at or below it. What a p-value isn't matters as much. In its 2016 statement on statistical significance and p-values, the American Statistical Association stressed that a p-value doesn't measure the probability that the hypothesis being studied is true, or that the data arose by chance alone, and that it doesn't measure the size of an effect or the importance of a result.
p = 0.03 means
- If the null were true, a result at least this extreme would turn up about 3% of the time.
- At α = 0.05, you reject the null.
- The data are hard to square with no difference.
p = 0.03 does not mean
- There is a 3% chance the null hypothesis is true.
- There is a 97% chance the alternative is true.
- The effect is large or important.
- The result would almost certainly hold up in a repeat study.
Because p says nothing about size, report the effect itself (the difference in means or rates) with a confidence interval alongside it. A reader can then judge whether the difference matters, not just whether it cleared 0.05.
Why you never accept the null
A test ends in one of two decisions: reject the null hypothesis, or fail to reject it. Penn State's review is explicit that not rejecting the null doesn't prove it true. Failing to reject means the data weren't strong enough to rule the null out, which happens when there is no effect, and also when the study was too small or too noisy to detect one.
Douglas Altman and Martin Bland put the point in the title of a 1995 statistics note in the BMJ, “Absence of evidence is not evidence of absence.” A trial reported as showing “no difference” has usually shown only that it found no evidence of one, and as they write, “we can never prove the absence of a relation.” NIST's handbook frames the decision practically: not rejecting can be a perfectly good result if you want to go on acting as if the null were true. It just isn't a finding that the null is true.
| The null is true | The null is false | |
|---|---|---|
| You reject the null | The null is trueType I error, a false positive. Its risk is α. | The null is falseCorrect decision |
| You fail to reject the null | The null is trueCorrect decision | The null is falseType II error, a false negative. Its risk is β. |
Showing that two groups are practically the same takes a different test. In an equivalence test you set in advance the smallest difference that would matter, say 5 percentage points, and the null becomes that the true difference is at least that large in either direction. Rejecting it, which in practice means the confidence interval for the difference falls entirely inside the margin, is evidence of equivalence. The tutoring result above doesn't come close: its interval runs to 10 points, so it shows no evidence of a difference, not evidence that there is none.
For how to carry a result like this into the last section of a paper, see the guide on writing a research paper conclusion; the research paper example shows hypotheses and results side by side in a full IMRaD paper.
Common null hypothesis mistakes
- Putting your prediction in the null. The null is the claim you test against. What you expect goes in the alternative.
- Writing hypotheses about the sample. H0 is about the population mean μ, not the sample mean you calculate from your data.
- Choosing the tail after seeing the results. Decide one- or two-tailed when you write the hypotheses, not when a two-tailed p comes back at 0.08.
- “Accepting” or “proving” the null. Write “failed to reject” or “found no significant difference,” and give the confidence interval.
- Reading p as the size of the effect. A tiny, unimportant difference can be highly significant in a large sample.
- Writing the ANOVA alternative as “all means differ.” The alternative is that at least one mean differs from the others.
Once the results section is drafted, you can ask the Strokes Assistant to review it and flag any sentence that accepts or proves the null, or reads a p-value as the chance a hypothesis is true.
Which test fits depends on your design as much as your hypotheses. The guide to qualitative vs quantitative research covers that choice, and the methodology section example shows how to describe the analysis.
Frequently asked questions
What does it mean to reject the null hypothesis?
It means your data would be unlikely if the null were true: the p-value came in at or below the significance level you set, such as 0.05, so you conclude the evidence supports the alternative. It doesn't prove the alternative, and it says nothing about how large or important the effect is. Report the effect size and a confidence interval too.
Can you accept the null hypothesis?
Not in standard hypothesis testing. The two possible decisions are to reject the null or to fail to reject it. Failing to reject means the evidence wasn't strong enough, which can happen simply because a study was too small. Showing that an effect is negligible takes an equivalence test, with a margin set in advance, not a non-significant result.
Is the null hypothesis always “no difference”?
Usually, but not always. A null can state a specific value, such as a mean bulb life of 500 hours or a mean rent of $1,450, or a predicted pattern, such as a 3:1 ratio of traits in a genetic cross. In a goodness-of-fit test the null is the theory's prediction, so researchers often hope the data won't reject it.
