Null and Alternative Hypotheses | Definitions & Examples

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test:

• Null hypothesis (H₀): There’s no effect in the population.
• Alternative hypothesis (H_A): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable.

Answering your research question with hypotheses

The null and alternative hypotheses offer competing answers to your research question. When the research question asks “Does the independent variable affect the dependent variable?”, the null hypothesis (H₀) answers “No, there’s no effect in the population.” On the other hand, the alternative hypothesis (H_A) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample. Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample.

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

What is a null hypothesis?

The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population (p ≤ α), then we can reject the null hypothesis. Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept. Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect”, “no difference”, or “no relationship”. When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p₁ = p₂.

What is an alternative hypothesis?

The alternative hypothesis (H_A) is the other answer to your research question. It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect”, “a difference”, or “a relationship”. When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes > or <). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Differences between null and alternative hypotheses

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question
• They both make claims about the population
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

How to write null and alternative hypotheses

To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable?

• Null hypothesis (H₀): Independent variable does not affect dependent variable.
• Alternative hypothesis (H_A): Independent variable affects dependent variable.

Test-specific

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests. One-tailed tests are appropriate for most studies.

Frequently asked questions about null and alternative hypotheses

What symbols are used to represent null hypotheses?

The null hypothesis is often abbreviated as H₀. When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

What symbols are used to represent alternative hypotheses?

The alternative hypothesis is often abbreviated as H_a or H₁. When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

What’s the difference between a research hypothesis and a statistical hypothesis?

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study, the statistical hypotheses correspond logically to the research hypothesis.