The null hypothesis
The null hypothesis is the statement that there is no effect, no difference and no relationship, and it is the statement a statistical test is actually built to examine. A test never sets out to prove the interesting claim. It assumes the boring one, works out how surprising the data would be if the boring one were true, and rejects it only when the data are surprising enough.
That backwards arrangement confuses people, and it is deliberate. "No difference" is a single, precise statement that fixes exactly what the data should look like. "There is a difference" is a whole family of statements, one for every possible size of difference, and a family cannot be tested. So the null gets the arithmetic, and the alternative gets whatever the null leaves behind.
What the null hypothesis is
The null hypothesis, written H nought, is the default assumption a test provisionally accepts so that it has something specific to calculate against. In a comparison of two production lines it says the two lines have the same fault rate. In a study of a training program it says the average score after training equals the average score before. In a test of association it says the two variables are independent.
What distinguishes the null hypothesis from an ordinary assumption is that it must be exact enough to generate a distribution. From "the two lines have the same fault rate" you can compute exactly how often a gap of any given size would occur by chance, which is what makes a p value possible. From "the new line is better" you cannot, because better by how much is unspecified. The null is chosen for computability, not for plausibility, and it is usually the hypothesis nobody believes.
The one-sentence version: the claim you are trying to knock down
In one sentence: the null hypothesis is the position you argue against, and evidence counts only when it would be unlikely under that position. Everything a significance test reports is a statement about how badly the data fit the null.
Two consequences follow immediately, and both are routinely dropped in reporting. First, rejecting the null tells you the data are unlikely under no effect; it does not tell you the size of the effect, nor that your favored explanation is the right one, only that this particular no-effect story fits badly. Second, not rejecting the null tells you the data were compatible with no effect; it does not tell you there is no effect. The second point is the one this page exists to make, and it is worked through below with numbers.
The arithmetic worked through: two production lines and a chi square test
Work the arithmetic on 400 circuit boards, 200 from each of two production lines, with 18 faults found on line A and 12 on line B. These are invented illustrative figures. Line A has 50 percent more faults than line B, which looks like a clear difference, and the test is going to disagree.
The null hypothesis here is that fault rate is independent of line: both lines share one underlying rate. Start by computing what that assumption predicts.
- Total faults: 18 plus 12, which is 30, out of 400 boards, a shared rate of 7.5 percent.
- Expected faults on each line under the null: 7.5 percent of 200, which is 15.
- Expected sound boards on each line: 200 minus 15, which is 185.
- Observed against expected: line A has 18 faults where 15 were expected, line B has 12 where 15 were expected. Each cell is off by 3.
The chi square statistic sums the squared gap divided by the expectation, across all four cells. The two fault cells each contribute 3 squared over 15, which is 9 divided by 15, or 0.600. The two sound cells each contribute 3 squared over 185, which is 9 divided by 185, or 0.049. Adding them gives 0.600 plus 0.600 plus 0.049 plus 0.049, so chi square equals 1.30 with 1 degree of freedom.
The critical value for 1 degree of freedom at the 0.05 level is 3.84, and 1.30 is well below it. The corresponding p value is about 0.25, meaning a gap this size or larger would appear about a quarter of the time if the two lines really were identical. The null hypothesis is not rejected. Note what that verdict does not license: line A still recorded 50 percent more faults, and the honest summary is that 400 boards cannot distinguish that gap from ordinary variation, not that the lines are the same.
What a failure looks like: no effect found, reported as no effect
A null hypothesis failure looks like the sentence "the study found no effect", written about a result that simply failed to reach a threshold. The two production lines above would generate exactly that headline. Everything in the arithmetic is correct and the conclusion is unsupported, because a test that fails to reject has told you about the sensitivity of the test as much as about the world.
The size of the sample decides which of those two things you are learning. Keep the same fault rates, 9 percent on line A and 6 percent on line B, and run 2,000 boards from each line instead of 200. Expected faults become 180 and 120 against a shared expectation of 150, each cell now off by 30 rather than 3, and the same test comfortably rejects the null. Same effect, same rates, different resolving power. Whenever a result is reported as showing no difference, the first question is how large a difference the design could have detected, which is a question about sample size.
The common misreading: absence of evidence
The common misreading is to treat failure to reject the null as proof that the null is true. It is not, and the language of the discipline is careful about this for a reason: results are described as failing to reject, never as accepting or proving. A test is a filter with a known false alarm rate and an often unknown miss rate, and passing through the filter is weak information.
Three more misreadings are worth naming, because they travel together.
- Rejecting the null does not confirm the alternative hypothesis you had in mind; it rules out one specific no-effect story, and other explanations may fit the data equally well.
- The null hypothesis is not the hypothesis the researcher believes; it is the one the arithmetic requires, and it is usually assumed only in order to be discarded.
- A p value above 0.05 does not mean the effect is zero, and reporting a range of effect sizes the data are compatible with is more informative than reporting the verdict.
There is a further trap in choosing the null badly. A null of "no difference at all" is easy to reject with a large enough sample, since almost nothing is exactly equal. Rejecting it then becomes a statement about sample size rather than about the world, which is the point where statistical significance and practical importance separate.
The test to run on any reported result
Run these five questions on any result that reports a test, whether it rejected the null or not.
- State the null hypothesis in one sentence, and check whether the report ever does.
- Ask what result would have counted as evidence against the researcher's own claim.
- Ask what size of effect the design could have detected, if the null was not rejected.
- Ask how many tests were run in total, since each one carries its own chance of a false rejection.
- Ask whether the effect size is reported alongside the verdict, in the units of the thing measured.
The second question is the one to keep. A claim whose author cannot say what evidence would count against it has no null hypothesis, and no test either. Converting the reported result back into raw counts before arguing about it is the habit taught in Reading numbers, the threshold that decides the verdict is covered under Statistical significance, and the gambler's fallacy shows what happens when the null is discarded on a run of results that chance produces routinely. Critical thinking exercises drill the habit of stating the null before reading the conclusion.
Null against alternative: symbols, wording and four standard tests
The alternative hypothesis, written H one or H a, is everything the null excludes, and the pair must be exhaustive and mutually exclusive. The null always contains the equality, whether written as equals, at most or at least; the alternative never does. The null is rejected when the p value falls below the significance level set in advance, and it is not rejected otherwise. There is no third verdict.
- Chi square test of independence: the null states that the two categorical variables are independent, so the observed cell counts differ from the expected counts only by chance.
- Chi square goodness of fit: the null states that the observed frequencies match a specified distribution, such as equal counts across six categories.
- Paired t test: the null states that the mean of the differences between paired measurements equals zero, which is a statement about the differences, not about the two sets of raw scores.
- Two-sample t test: the null states that the two population means are equal, with the alternative left one-sided or two-sided depending on what was specified before the data were seen.
The wording differs and the logic does not. Each test computes how far the data sit from what the null predicts, converts that distance into a probability, and compares it to a threshold. The honest limit of the whole apparatus is that it can only ever reject a hypothesis nobody held, and it can never confirm the one somebody wanted.
What a null hypothesis calculator asks you for, and what it never asks
No calculator computes a null hypothesis, and the closest thing to one is worth looking at carefully for exactly that reason. Chi square calculators, t test calculators and p value converters all take a test statistic, or the raw counts to build one, plus the degrees of freedom, and return a probability. Feed the two production lines from the calculation above into a chi square tool and it returns 1.30 with 1 degree of freedom and a p value of about 0.25. Feed it the 2,000-board version and it returns about 13.0 and a p value near 0.0003. Both answers are correct and neither required the tool to know what was being tested.
Look at the input form of any such calculator and notice what is not on it. There is no field for the null hypothesis. There is no field for what the numbers represent, whether the comparison was planned before the data arrived, how many other comparisons were run, or whether the two groups differ in some way other than the one being tested. The calculator asks for a table and a distribution, and those two inputs already encode a null hypothesis that somebody chose. A chi square test of independence and a chi square goodness of fit test can be run on the same counts and are testing different claims, so the same table can produce different verdicts depending on a decision the tool never mentions.
What the output does not settle is the direction the whole page has been arguing. A p value of 0.25 does not mean the two lines are the same; it means 400 boards could not distinguish them, and the calculator has no field for how large a difference the design could have detected. A p value of 0.0003 does not mean the difference matters; it means the difference is hard to attribute to chance at that sample size. Before you run the numbers, write down the null hypothesis in one sentence and the threshold you will judge it against. If you cannot state the null, the calculator will still return a number, and the number will be an answer to a question nobody asked.