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Statistics By Jim

Making statistics intuitive

Null Hypothesis: Definition, Rejecting & Examples

By Jim Frost 6 Comments

What is a Null Hypothesis?

The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

• Null Hypothesis H 0 : No effect exists in the population.
• Alternative Hypothesis H A : The effect exists in the population.

In every study or experiment, researchers assess an effect or relationship. This effect can be the effectiveness of a new drug, building material, or other intervention that has benefits. There is a benefit or connection that the researchers hope to identify. Unfortunately, no effect may exist. In statistics, we call this lack of an effect the null hypothesis. Researchers assume that this notion of no effect is correct until they have enough evidence to suggest otherwise, similar to how a trial presumes innocence.

In this context, the analysts don’t necessarily believe the null hypothesis is correct. In fact, they typically want to reject it because that leads to more exciting finds about an effect or relationship. The new vaccine works!

You can think of it as the default theory that requires sufficiently strong evidence to reject. Like a prosecutor, researchers must collect sufficient evidence to overturn the presumption of no effect. Investigators must work hard to set up a study and a data collection system to obtain evidence that can reject the null hypothesis.

Related post : What is an Effect in Statistics?

Null Hypothesis Examples

Null hypotheses start as research questions that the investigator rephrases as a statement indicating there is no effect or relationship.

After reading these examples, you might think they’re a bit boring and pointless. However, the key is to remember that the null hypothesis defines the condition that the researchers need to discredit before suggesting an effect exists.

Let’s see how you reject the null hypothesis and get to those more exciting findings!

When to Reject the Null Hypothesis

So, you want to reject the null hypothesis, but how and when can you do that? To start, you’ll need to perform a statistical test on your data. The following is an overview of performing a study that uses a hypothesis test.

The first step is to devise a research question and the appropriate null hypothesis. After that, the investigators need to formulate an experimental design and data collection procedures that will allow them to gather data that can answer the research question. Then they collect the data. For more information about designing a scientific study that uses statistics, read my post 5 Steps for Conducting Studies with Statistics .

After data collection is complete, statistics and hypothesis testing enter the picture. Hypothesis testing takes your sample data and evaluates how consistent they are with the null hypothesis. The p-value is a crucial part of the statistical results because it quantifies how strongly the sample data contradict the null hypothesis.

When the sample data provide sufficient evidence, you can reject the null hypothesis. In a hypothesis test, this process involves comparing the p-value to your significance level .

Rejecting the Null Hypothesis

Reject the null hypothesis when the p-value is less than or equal to your significance level. Your sample data favor the alternative hypothesis, which suggests that the effect exists in the population. For a mnemonic device, remember—when the p-value is low, the null must go!

When you can reject the null hypothesis, your results are statistically significant. Learn more about Statistical Significance: Definition & Meaning .

Failing to Reject the Null Hypothesis

Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis. The sample data provides insufficient data to conclude that the effect exists in the population. When the p-value is high, the null must fly!

Note that failing to reject the null is not the same as proving it. For more information about the difference, read my post about Failing to Reject the Null .

That’s a very general look at the process. But I hope you can see how the path to more exciting findings depends on being able to rule out the less exciting null hypothesis that states there’s nothing to see here!

Let’s move on to learning how to write the null hypothesis for different types of effects, relationships, and tests.

Related posts : How Hypothesis Tests Work and Interpreting P-values

How to Write a Null Hypothesis

The null hypothesis varies by the type of statistic and hypothesis test. Remember that inferential statistics use samples to draw conclusions about populations. Consequently, when you write a null hypothesis, it must make a claim about the relevant population parameter . Further, that claim usually indicates that the effect does not exist in the population. Below are typical examples of writing a null hypothesis for various parameters and hypothesis tests.

Related posts : Descriptive vs. Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

Group Means

T-tests and ANOVA assess the differences between group means. For these tests, the null hypothesis states that there is no difference between group means in the population. In other words, the experimental conditions that define the groups do not affect the mean outcome. Mu (µ) is the population parameter for the mean, and you’ll need to include it in the statement for this type of study.

For example, an experiment compares the mean bone density changes for a new osteoporosis medication. The control group does not receive the medicine, while the treatment group does. The null states that the mean bone density changes for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group means are equal in the population: µ 1 = µ 2 , or µ 1 – µ 2 = 0
• Alternative Hypothesis H A : Group means are not equal in the population: µ 1 ≠ µ 2 , or µ 1 – µ 2 ≠ 0.

Group Proportions

Proportions tests assess the differences between group proportions. For these tests, the null hypothesis states that there is no difference between group proportions. Again, the experimental conditions did not affect the proportion of events in the groups. P is the population proportion parameter that you’ll need to include.

For example, a vaccine experiment compares the infection rate in the treatment group to the control group. The treatment group receives the vaccine, while the control group does not. The null states that the infection rates for the control and treatment groups are equal.

• Null Hypothesis H 0 : Group proportions are equal in the population: p 1 = p 2 .
• Alternative Hypothesis H A : Group proportions are not equal in the population: p 1 ≠ p 2 .

Correlation and Regression Coefficients

Some studies assess the relationship between two continuous variables rather than differences between groups.

In these studies, analysts often use either correlation or regression analysis . For these tests, the null states that there is no relationship between the variables. Specifically, it says that the correlation or regression coefficient is zero. As one variable increases, there is no tendency for the other variable to increase or decrease. Rho (ρ) is the population correlation parameter and beta (β) is the regression coefficient parameter.

For example, a study assesses the relationship between screen time and test performance. The null states that there is no correlation between this pair of variables. As screen time increases, test performance does not tend to increase or decrease.

• Null Hypothesis H 0 : The correlation in the population is zero: ρ = 0.
• Alternative Hypothesis H A : The correlation in the population is not zero: ρ ≠ 0.

For all these cases, the analysts define the hypotheses before the study. After collecting the data, they perform a hypothesis test to determine whether they can reject the null hypothesis.

The preceding examples are all for two-tailed hypothesis tests. To learn about one-tailed tests and how to write a null hypothesis for them, read my post One-Tailed vs. Two-Tailed Tests .

Related post : Understanding Correlation

Neyman, J; Pearson, E. S. (January 1, 1933).  On the Problem of the most Efficient Tests of Statistical Hypotheses .  Philosophical Transactions of the Royal Society A .  231  (694–706): 289–337.

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January 11, 2024 at 2:57 pm

Thanks for the reply.

January 10, 2024 at 1:23 pm

Hi Jim, In your comment you state that equivalence test null and alternate hypotheses are reversed. For hypothesis tests of data fits to a probability distribution, the null hypothesis is that the probability distribution fits the data. Is this correct?

January 10, 2024 at 2:15 pm

Those two separate things, equivalence testing and normality tests. But, yes, you’re correct for both.

Hypotheses are switched for equivalence testing. You need to “work” (i.e., collect a large sample of good quality data) to be able to reject the null that the groups are different to be able to conclude they’re the same.

With typical hypothesis tests, if you have low quality data and a low sample size, you’ll fail to reject the null that they’re the same, concluding they’re equivalent. But that’s more a statement about the low quality and small sample size than anything to do with the groups being equal.

So, equivalence testing make you work to obtain a finding that the groups are the same (at least within some amount you define as a trivial difference).

For normality testing, and other distribution tests, the null states that the data follow the distribution (normal or whatever). If you reject the null, you have sufficient evidence to conclude that your sample data don’t follow the probability distribution. That’s a rare case where you hope to fail to reject the null. And it suffers from the problem I describe above where you might fail to reject the null simply because you have a small sample size. In that case, you’d conclude the data follow the probability distribution but it’s more that you don’t have enough data for the test to register the deviation. In this scenario, if you had a larger sample size, you’d reject the null and conclude it doesn’t follow that distribution.

I don’t know of any equivalence testing type approach for distribution fit tests where you’d need to work to show the data follow a distribution, although I haven’t looked for one either!

February 20, 2022 at 9:26 pm

Is a null hypothesis regularly (always) stated in the negative? “there is no” or “does not”

February 23, 2022 at 9:21 pm

Typically, the null hypothesis includes an equal sign. The null hypothesis states that the population parameter equals a particular value. That value is usually one that represents no effect. In the case of a one-sided hypothesis test, the null still contains an equal sign but it’s “greater than or equal to” or “less than or equal to.” If you wanted to translate the null hypothesis from its native mathematical expression, you could use the expression “there is no effect.” But the mathematical form more specifically states what it’s testing.

It’s the alternative hypothesis that typically contains does not equal.

There are some exceptions. For example, in an equivalence test where the researchers want to show that two things are equal, the null hypothesis states that they’re not equal.

In short, the null hypothesis states the condition that the researchers hope to reject. They need to work hard to set up an experiment and data collection that’ll gather enough evidence to be able to reject the null condition.

February 15, 2022 at 9:32 am

Dear sir I always read your notes on Research methods.. Kindly tell is there any available Book on all these..wonderfull Urgent

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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

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

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

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 ( H 0 ) answers “No, there’s no effect in the population.”
• 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. It’s critical for your research to write strong hypotheses .

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.

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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 ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

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 p 1 = p 2 .

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.

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.

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 template sentences

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 ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

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.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

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.

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What is The Null Hypothesis & When Do You Reject The Null Hypothesis

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Editor at Simply Psychology

BA (Hons) Psychology, Princeton University

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On This Page:

A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating ​ the dependent variable or due to random chance. 

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

When Do We Reject The Null Hypothesis? 

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected. 

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables. 

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a  p  -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes be incorrect.

When you incorrectly reject the null hypothesis, it’s called a type I error. When you incorrectly fail to reject it, it’s called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. 

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null. 

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists. 

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter. 

Purpose of a Null Hypothesis 

• The primary purpose of the null hypothesis is to disprove an assumption. 
• Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
• A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true. 

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables. 

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study. 

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”).  However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

Gill, J. (1999). The insignificance of null hypothesis significance testing.  Political research quarterly ,  52 (3), 647-674.

Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method.  American Psychologist ,  56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing.  Behavior research methods ,  43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy.  Psychological methods ,  5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test.  Psychological bulletin ,  57 (5), 416.

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null hypothesis

Definition of null hypothesis

Examples of null hypothesis in a sentence.

These examples are programmatically compiled from various online sources to illustrate current usage of the word 'null hypothesis.' Any opinions expressed in the examples do not represent those of Merriam-Webster or its editors. Send us feedback about these examples.

Word History

1935, in the meaning defined above

Dictionary Entries Near null hypothesis

Nullarbor Plain

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“Null hypothesis.” Merriam-Webster.com Dictionary , Merriam-Webster, https://www.merriam-webster.com/dictionary/null%20hypothesis. Accessed 24 Jun. 2024.

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9.1: Null and Alternative Hypotheses

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The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

\(H_0\): The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

\(H_a\): The alternative hypothesis: It is a claim about the population that is contradictory to \(H_0\) and what we conclude when we reject \(H_0\). This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject \(H_0\)" if the sample information favors the alternative hypothesis or "do not reject \(H_0\)" or "decline to reject \(H_0\)" if the sample information is insufficient to reject the null hypothesis.

\(H_{0}\) always has a symbol with an equal in it. \(H_{a}\) never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example \(\PageIndex{1}\)

• \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
• \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

• \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
• \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are:

• \(H_{0}: \mu = 2.0\)
• \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 66\)
• \(H_{a}: \mu \_ 66\)
• \(H_{0}: \mu = 66\)
• \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are:

• \(H_{0}: \mu \geq 5\)
• \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• \(H_{0}: \mu \_ 45\)
• \(H_{a}: \mu \_ 45\)
• \(H_{0}: \mu \geq 45\)
• \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses.

• \(H_{0}: p \leq 0.066\)
• \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

• \(H_{0}: p \_ 0.40\)
• \(H_{a}: p \_ 0.40\)
• \(H_{0}: p = 0.40\)
• \(H_{a}: p > 0.40\)

COLLABORATIVE EXERCISE

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

• Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
• Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
• If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
• Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

• If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
• If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

Null Hypothesis Definition and Examples, How to State

What is the null hypothesis, how to state the null hypothesis, null hypothesis overview.

Why is it Called the “Null”?

The word “null” in this context means that it’s a commonly accepted fact that researchers work to nullify . It doesn’t mean that the statement is null (i.e. amounts to nothing) itself! (Perhaps the term should be called the “nullifiable hypothesis” as that might cause less confusion).

Why Do I need to Test it? Why not just prove an alternate one?

The short answer is, as a scientist, you are required to ; It’s part of the scientific process. Science uses a battery of processes to prove or disprove theories, making sure than any new hypothesis has no flaws. Including both a null and an alternate hypothesis is one safeguard to ensure your research isn’t flawed. Not including the null hypothesis in your research is considered very bad practice by the scientific community. If you set out to prove an alternate hypothesis without considering it, you are likely setting yourself up for failure. At a minimum, your experiment will likely not be taken seriously.

• Null hypothesis : H 0 : The world is flat.
• Alternate hypothesis: The world is round.

Several scientists, including Copernicus , set out to disprove the null hypothesis. This eventually led to the rejection of the null and the acceptance of the alternate. Most people accepted it — the ones that didn’t created the Flat Earth Society !. What would have happened if Copernicus had not disproved the it and merely proved the alternate? No one would have listened to him. In order to change people’s thinking, he first had to prove that their thinking was wrong .

How to State the Null Hypothesis from a Word Problem

You’ll be asked to convert a word problem into a hypothesis statement in statistics that will include a null hypothesis and an alternate hypothesis . Breaking your problem into a few small steps makes these problems much easier to handle.

Step 2: Convert the hypothesis to math . Remember that the average is sometimes written as μ.

H 1 : μ > 8.2

Broken down into (somewhat) English, that’s H 1 (The hypothesis): μ (the average) > (is greater than) 8.2

Step 3: State what will happen if the hypothesis doesn’t come true. If the recovery time isn’t greater than 8.2 weeks, there are only two possibilities, that the recovery time is equal to 8.2 weeks or less than 8.2 weeks.

H 0 : μ ≤ 8.2

Broken down again into English, that’s H 0 (The null hypothesis): μ (the average) ≤ (is less than or equal to) 8.2

How to State the Null Hypothesis: Part Two

But what if the researcher doesn’t have any idea what will happen.

Example Problem: A researcher is studying the effects of radical exercise program on knee surgery patients. There is a good chance the therapy will improve recovery time, but there’s also the possibility it will make it worse. Average recovery times for knee surgery patients is 8.2 weeks. 

Step 1: State what will happen if the experiment doesn’t make any difference. That’s the null hypothesis–that nothing will happen. In this experiment, if nothing happens, then the recovery time will stay at 8.2 weeks.

H 0 : μ = 8.2

Broken down into English, that’s H 0 (The null hypothesis): μ (the average) = (is equal to) 8.2

Step 2: Figure out the alternate hypothesis . The alternate hypothesis is the opposite of the null hypothesis. In other words, what happens if our experiment makes a difference?

H 1 : μ ≠ 8.2

In English again, that’s H 1 (The  alternate hypothesis): μ (the average) ≠ (is not equal to) 8.2

That’s How to State the Null Hypothesis!

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Gonick, L. (1993). The Cartoon Guide to Statistics . HarperPerennial. Kotz, S.; et al., eds. (2006), Encyclopedia of Statistical Sciences , Wiley.

9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 , the — null hypothesis: a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a —, the alternative hypothesis: a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30 H a : More than 30 percent of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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• Null hypothesis

by Marco Taboga , PhD

In a test of hypothesis , a sample of data is used to decide whether to reject or not to reject a hypothesis about the probability distribution from which the sample was extracted.

The hypothesis is called the null hypothesis, or simply "the null".

Table of contents

The null is like the defendant in a criminal trial

How is the null hypothesis tested, example 1 - proportion of defective items, measurement, test statistic, critical region, interpretation, example 2 - reliability of a production plant, rejection and failure to reject, not rejecting and accepting are not the same thing, failure to reject can be due to lack of power, rejections are easier to interpret, but be careful, takeaways - how to (and not to) formulate a null hypothesis, more examples, more details, best practices in science, keep reading the glossary.

Formulating null hypotheses and subjecting them to statistical testing is one of the workhorses of the scientific method.

Scientists in all fields make conjectures about the phenomena they study, translate them into null hypotheses and gather data to test them.

This process resembles a trial:

the defendant (the null hypothesis) is accused of being guilty (wrong);

evidence (data) is gathered in order to prove the defendant guilty (reject the null);

if there is evidence beyond any reasonable doubt, the defendant is found guilty (the null is rejected);

otherwise, the defendant is found not guilty (the null is not rejected).

Keep this analogy in mind because it helps to better understand statistical tests, their limitations, use and misuse, and frequent misinterpretation.

Before collecting the data:

we decide how to summarize the relevant characteristics of the sample data in a single number, the so-called test statistic ;

we derive the probability distribution of the test statistic under the hypothesis that the null is true (the data is regarded as random; therefore, the test statistic is a random variable);

we decide what probability of incorrectly rejecting the null we are willing to tolerate (the level of significance , or size of the test ); the level of significance is typically a small number, such as 5% or 1%.

we choose one or more intervals of values (collectively called rejection region) such that the probability that the test statistic falls within these intervals is equal to the desired level of significance; the rejection region is often a tail of the distribution of the test statistic (one-tailed test) or the union of the left and right tails (two-tailed test).

Then, the data is collected and used to compute the value of the test statistic.

A decision is taken as follows:

if the test statistic falls within the rejection region, then the null hypothesis is rejected;

otherwise, it is not rejected.

We now make two examples of practical problems that lead to formulate and test a null hypothesis.

A new method is proposed to produce light bulbs.

The proponents claim that it produces less defective bulbs than the method currently in use.

To check the claim, we can set up a statistical test as follows.

We keep the light bulbs on for 10 consecutive days, and then we record whether they are still working at the end of the test period.

The probability that a light bulb produced with the new method is still working at the end of the test period is the same as that of a light bulb produced with the old method.

100 light bulbs are tested:

50 of them are produced with the new method (group A)

the remaining 50 are produced with the old method (group B).

The final data comprises 100 observations of:

an indicator variable which is equal to 1 if the light bulb is still working at the end of the test period and 0 otherwise;

a categorical variable that records the group (A or B) to which each light bulb belongs.

We use the data to compute the proportions of working light bulbs in groups A and B.

The proportions are estimates of the probabilities of not being defective, which are equal for the two groups under the null hypothesis.

We then compute a z-statistic (see here for details) by:

taking the difference between the proportion in group A and the proportion in group B;

standardizing the difference:

we subtract the expected value (which is zero under the null hypothesis);

we divide by the standard deviation (it can be derived analytically).

The distribution of the z-statistic can be approximated by a standard normal distribution .

We decide that the level of confidence must be 5%. In other words, we are going to tolerate a 5% probability of incorrectly rejecting the null hypothesis.

The critical region is the right 5%-tail of the normal distribution, that is, the set of all values greater than 1.645 (see the glossary entry on critical values if you are wondering how this value was obtained).

If the test statistic is greater than 1.645, then the null hypothesis is rejected; otherwise, it is not rejected.

A rejection is interpreted as significant evidence that the new production method produces less defective items; failure to reject is interpreted as insufficient evidence that the new method is better.

A production plant incurs high costs when production needs to be halted because some machinery fails.

The plant manager has decided that he is not willing to tolerate more than one halt per year on average.

If the expected number of halts per year is greater than 1, he will make new investments in order to improve the reliability of the plant.

A statistical test is set up as follows.

The reliability of the plant is measured by the number of halts.

The number of halts in a year is assumed to have a Poisson distribution with expected value equal to 1 (using the Poisson distribution is common in reliability testing).

The manager cannot wait more than one year before taking a decision.

There will be a single datum at his disposal: the number of halts observed during one year.

The number of halts is used as a test statistic. By assumption, it has a Poisson distribution under the null hypothesis.

The manager decides that the probability of incorrectly rejecting the null can be at most 10%.

A Poisson random variable with expected value equal to 1 takes values:

larger than 1 with probability 26.42%;

larger than 2 with probability 8.03%.

Therefore, it is decided that the critical region will be the set of all values greater than or equal to 3.

If the test statistic is strictly greater than or equal to 3, then the null is rejected; otherwise, it is not rejected.

A rejection is interpreted as significant evidence that the production plant is not reliable enough (the average number of halts per year is significantly larger than tolerated).

Failure to reject is interpreted as insufficient evidence that the plant is unreliable.

This section discusses the main problems that arise in the interpretation of the outcome of a statistical test (reject / not reject).

When the test statistic does not fall within the critical region, then we do not reject the null hypothesis.

Does this mean that we accept the null? Not really.

In general, failure to reject does not constitute, per se, strong evidence that the null hypothesis is true .

Remember the analogy between hypothesis testing and a criminal trial. In a trial, when the defendant is declared not guilty, this does not mean that the defendant is innocent. It only means that there was not enough evidence (not beyond any reasonable doubt) against the defendant.

In turn, lack of evidence can be due:

either to the fact that the defendant is innocent ;

or to the fact that the prosecution has not been able to provide enough evidence against the defendant, even if the latter is guilty .

This is the very reason why courts do not declare defendants innocent, but they use the locution "not guilty".

In a similar fashion, statisticians do not say that the null hypothesis has been accepted, but they say that it has not been rejected.

To better understand why failure to reject does not in general constitute strong evidence that the null hypothesis is true, we need to use the concept of statistical power .

The power of a test is the probability (calculated ex-ante, i.e., before observing the data) that the null will be rejected when another hypothesis (called the alternative hypothesis ) is true.

Let's consider the first of the two examples above (the production of light bulbs).

In that example, the null hypothesis is: the probability that a light bulb is defective does not decrease after introducing a new production method.

Let's make the alternative hypothesis that the probability of being defective is 1% smaller after changing the production process (assume that a 1% decrease is considered a meaningful improvement by engineers).

How much is the ex-ante probability of rejecting the null if the alternative hypothesis is true?

If this probability (the power of the test) is small, then it is very likely that we will not reject the null even if it is wrong.

If we use the analogy with criminal trials, low power means that most likely the prosecution will not be able to provide sufficient evidence, even if the defendant is guilty.

Thus, in the case of lack of power, failure to reject is almost meaningless (it was anyway highly likely).

This is why, before performing a test, it is good statistical practice to compute its power against a relevant alternative .

If the power is found to be too small, there are usually remedies. In particular, statistical power can usually be increased by increasing the sample size (see, e.g., the lecture on hypothesis tests about the mean ).

As we have explained above, interpreting a failure to reject the null hypothesis is not always straightforward. Instead, interpreting a rejection is somewhat easier.

When we reject the null, we know that the data has provided a lot of evidence against the null. In other words, it is unlikely (how unlikely depends on the size of the test) that the null is true given the data we have observed.

There is an important caveat though. The null hypothesis is often made up of several assumptions, including:

the main assumption (the one we are testing);

other assumptions (e.g., technical assumptions) that we need to make in order to set up the hypothesis test.

For instance, in Example 2 above (reliability of a production plant), the main assumption is that the expected number of production halts per year is equal to 1. But there is also a technical assumption: the number of production halts has a Poisson distribution.

It must be kept in mind that a rejection is always a joint rejection of the main assumption and all the other assumptions .

Therefore, we should always ask ourselves whether the null has been rejected because the main assumption is wrong or because the other assumptions are violated.

In the case of Example 2 above, is a rejection of the null due to the fact that the expected number of halts is greater than 1 or is it due to the fact that the distribution of the number of halts is very different from a Poisson distribution?

When we suspect that a rejection is due to the inappropriateness of some technical assumption (e.g., assuming a Poisson distribution in the example), we say that the rejection could be due to misspecification of the model .

The right thing to do when these kind of suspicions arise is to conduct so-called robustness checks , that is, to change the technical assumptions and carry out the test again.

In our example, we could re-run the test by assuming a different probability distribution for the number of halts (e.g., a negative binomial or a compound Poisson - do not worry if you have never heard about these distributions).

If we keep obtaining a rejection of the null even after changing the technical assumptions several times, the we say that our rejection is robust to several different specifications of the model .

What are the main practical implications of everything we have said thus far? How does the theory above help us to set up and test a null hypothesis?

What we said can be summarized in the following guiding principles:

A test of hypothesis is like a criminal trial and you are the prosecutor . You want to find evidence that the defendant (the null hypothesis) is guilty. Your job is not to prove that the defendant is innocent. If you find yourself hoping that the defendant is found not guilty (i.e., the null is not rejected) then something is wrong with the way you set up the test. Remember: you are the prosecutor.

Compute the power of your test against one or more relevant alternative hypotheses. Do not run a test if you know ex-ante that it is unlikely to reject the null when the alternative hypothesis is true.

Beware of technical assumptions that you add to the main assumption you want to test. Make robustness checks in order to verify that the outcome of the test is not biased by model misspecification.

More examples of null hypotheses and how to test them can be found in the following lectures.

The lecture on Hypothesis testing provides a more detailed mathematical treatment of null hypotheses and how they are tested.

This lecture on the null hypothesis was featured in Stanford University's Best practices in science .

Previous entry: Normal equations

Next entry: Parameter

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Taboga, Marco (2021). "Null hypothesis", Lectures on probability theory and mathematical statistics. Kindle Direct Publishing. Online appendix. https://www.statlect.com/glossary/null-hypothesis.

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Statistics Made Easy

How to Write a Null Hypothesis (5 Examples)

A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true.

Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

H 0 (Null Hypothesis): Population parameter =,  ≤, ≥ some value

H A  (Alternative Hypothesis): Population parameter <, >, ≠ some value

Note that the null hypothesis always contains the equal sign .

We interpret the hypotheses as follows:

Null hypothesis: The sample data provides no evidence to support some claim being made by an individual.

Alternative hypothesis: The sample data  does provide sufficient evidence to support the claim being made by an individual.

For example, suppose it’s assumed that the average height of a certain species of plant is 20 inches tall. However, one botanist claims the true average height is greater than 20 inches.

To test this claim, she may go out and collect a random sample of plants. She can then use this sample data to perform a hypothesis test using the following two hypotheses:

H 0 : μ ≤ 20 (the true mean height of plants is equal to or even less than 20 inches)

H A : μ > 20 (the true mean height of plants is greater than 20 inches)

If the sample data gathered by the botanist shows that the mean height of this species of plants is significantly greater than 20 inches, she can reject the null hypothesis and conclude that the mean height is greater than 20 inches.

Read through the following examples to gain a better understanding of how to write a null hypothesis in different situations.

Example 1: Weight of Turtles

A biologist wants to test whether or not the true mean weight of a certain species of turtles is 300 pounds. To test this, he goes out and measures the weight of a random sample of 40 turtles.

Here is how to write the null and alternative hypotheses for this scenario:

H 0 : μ = 300 (the true mean weight is equal to 300 pounds)

H A : μ ≠ 300 (the true mean weight is not equal to 300 pounds)

Example 2: Height of Males

It’s assumed that the mean height of males in a certain city is 68 inches. However, an independent researcher believes the true mean height is greater than 68 inches. To test this, he goes out and collects the height of 50 males in the city.

H 0 : μ ≤ 68 (the true mean height is equal to or even less than 68 inches)

H A : μ > 68 (the true mean height is greater than 68 inches)

Example 3: Graduation Rates

A university states that 80% of all students graduate on time. However, an independent researcher believes that less than 80% of all students graduate on time. To test this, she collects data on the proportion of students who graduated on time last year at the university.

H 0 : p ≥ 0.80 (the true proportion of students who graduate on time is 80% or higher)

H A : μ < 0.80 (the true proportion of students who graduate on time is less than 80%)

Example 4: Burger Weights

A food researcher wants to test whether or not the true mean weight of a burger at a certain restaurant is 7 ounces. To test this, he goes out and measures the weight of a random sample of 20 burgers from this restaurant.

H 0 : μ = 7 (the true mean weight is equal to 7 ounces)

H A : μ ≠ 7 (the true mean weight is not equal to 7 ounces)

Example 5: Citizen Support

A politician claims that less than 30% of citizens in a certain town support a certain law. To test this, he goes out and surveys 200 citizens on whether or not they support the law.

H 0 : p ≥ .30 (the true proportion of citizens who support the law is greater than or equal to 30%)

H A : μ < 0.30 (the true proportion of citizens who support the law is less than 30%)

Additional Resources

Introduction to Hypothesis Testing Introduction to Confidence Intervals An Explanation of P-Values and Statistical Significance

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

2 Replies to “How to Write a Null Hypothesis (5 Examples)”

you are amazing, thank you so much

Say I am a botanist hypothesizing the average height of daisies is 20 inches, or not? Does T = (ave – 20 inches) / √ variance / (80 / 4)? … This assumes 40 real measures + 40 fake = 80 n, but that seems questionable. Please advise.

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Null Hypothesis Examples

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In statistical analysis, the null hypothesis assumes there is no meaningful relationship between two variables. Testing the null hypothesis can tell you whether your results are due to the effect of manipulating ​a dependent variable or due to chance. It's often used in conjunction with an alternative hypothesis, which assumes there is, in fact, a relationship between two variables.

The null hypothesis is among the easiest hypothesis to test using statistical analysis, making it perhaps the most valuable hypothesis for the scientific method. By evaluating a null hypothesis in addition to another hypothesis, researchers can support their conclusions with a higher level of confidence. Below are examples of how you might formulate a null hypothesis to fit certain questions.

What Is the Null Hypothesis?

The null hypothesis states there is no relationship between the measured phenomenon (the dependent variable ) and the independent variable , which is the variable an experimenter typically controls or changes. You do not​ need to believe that the null hypothesis is true to test it. On the contrary, you will likely suspect there is a relationship between a set of variables. One way to prove that this is the case is to reject the null hypothesis. Rejecting a hypothesis does not mean an experiment was "bad" or that it didn't produce results. In fact, it is often one of the first steps toward further inquiry.

To distinguish it from other hypotheses , the null hypothesis is written as ​ H 0  (which is read as “H-nought,” "H-null," or "H-zero"). A significance test is used to determine the likelihood that the results supporting the null hypothesis are not due to chance. A confidence level of 95% or 99% is common. Keep in mind, even if the confidence level is high, there is still a small chance the null hypothesis is not true, perhaps because the experimenter did not account for a critical factor or because of chance. This is one reason why it's important to repeat experiments.

Examples of the Null Hypothesis

To write a null hypothesis, first start by asking a question. Rephrase that question in a form that assumes no relationship between the variables. In other words, assume a treatment has no effect. Write your hypothesis in a way that reflects this.

Other Types of Hypotheses

In addition to the null hypothesis, the alternative hypothesis is also a staple in traditional significance tests . It's essentially the opposite of the null hypothesis because it assumes the claim in question is true. For the first item in the table above, for example, an alternative hypothesis might be "Age does have an effect on mathematical ability."

Key Takeaways

• In hypothesis testing, the null hypothesis assumes no relationship between two variables, providing a baseline for statistical analysis.
• Rejecting the null hypothesis suggests there is evidence of a relationship between variables.
• By formulating a null hypothesis, researchers can systematically test assumptions and draw more reliable conclusions from their experiments.
• Difference Between Independent and Dependent Variables
• Examples of Independent and Dependent Variables
• What Is a Hypothesis? (Science)
• What 'Fail to Reject' Means in a Hypothesis Test
• Definition of a Hypothesis
• Null Hypothesis Definition and Examples
• Scientific Method Vocabulary Terms
• Null Hypothesis and Alternative Hypothesis
• Hypothesis Test for the Difference of Two Population Proportions
• How to Conduct a Hypothesis Test
• What Is a P-Value?
• What Are the Elements of a Good Hypothesis?
• Hypothesis Test Example
• What Is the Difference Between Alpha and P-Values?
• Understanding Path Analysis
• An Example of a Hypothesis Test

Null Hypothesis

Null hypothesis is used to make decisions based on data and by using statistical tests. Null hypothesis is represented using H o and it states that there is no difference between the characteristics of two samples. Null hypothesis is generally a statement of no difference. The rejection of null hypothesis is equivalent to the acceptance of the alternate hypothesis.

Let us learn more about null hypotheses, tests for null hypotheses, the difference between null hypothesis and alternate hypothesis, with the help of examples, FAQs.

What Is Null Hypothesis?

Null hypothesis states that there is no significant difference between the observed characteristics across two sample sets. Null hypothesis states the observed population parameters or variables is the same across the samples. The null hypothesis states that there is no relationship between the sample parameters, the independent variable, and the dependent variable. The term null hypothesis is used in instances to mean that there is no differences in the two means, or that the difference is not so significant.

If the experimental outcome is the same as the theoretical outcome then the null hypothesis holds good. But if there are any differences in the observed parameters across the samples then the null hypothesis is rejected, and we consider an alternate hypothesis. The rejection of the null hypothesis does not mean that there were flaws in the basic experimentation, but it sets the stage for further research. Generally, the strength of the evidence is tested against the null hypothesis.

Null hypothesis and alternate hypothesis are the two approaches used across statistics. The alternate hypothesis states that there is a significant difference between the parameters across the samples. The alternate hypothesis is the inverse of null hypothesis. An important reason to reject the null hypothesis and consider the alternate hypothesis is due to experimental or sampling errors.

Tests For Null Hypothesis

The two important approaches of statistical interference of null hypothesis are significance testing and hypothesis testing. The null hypothesis is a theoretical hypothesis and is based on insufficient evidence, which requires further testing to prove if it is true or false.

Significance Testing

The aim of significance testing is to provide evidence to reject the null hypothesis. If the difference is strong enough then reject the null hypothesis and accept the alternate hypothesis. The testing is designed to test the strength of the evidence against the hypothesis. The four important steps of significance testing are as follows.

• First state the null and alternate hypotheses.
• Calculate the test statistics.
• Find the p-value.
• Test the p-value with the α and decide if the null hypothesis should be rejected or accepted.

If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.

• Hypothesis Testing

Hypothesis testing takes the parameters from the sample and makes a derivation about the population. A hypothesis is an educated guess about a sample, which can be tested either through an experiment or an observation. Initially, a tentative assumption is made about the sample in the form of a null hypothesis.

There are four steps to perform hypothesis testing. They are:

• Identify the null hypothesis.
• Define the null hypothesis statement.
• Choose the test to be performed.
• Accept the null hypothesis or the alternate hypothesis.

There are often errors in the process of testing the hypothesis. The two important errors observed in hypothesis testing is as follows.

• Type - I error is rejecting the null hypothesis when the null hypothesis is actually true.
• Type - II error is accepting the null hypothesis when the null hypothesis is actually false.

Difference Between Null Hypothesis And Alternate Hypothesis

The difference between null hypothesis and alternate hypothesis can be understood through the following points.

• The opposite of the null hypothesis is the alternate hypothesis and it is the claim which is being proved by research to be true.
• The null hypothesis states that the two samples of the population are the same, and the alternate hypothesis states that there is a significant difference between the two samples of the population.
• The null hypothesis is designated as H o and the alternate hypothesis is designated as H a .
• For the null hypothesis, the same means are assumed to be equal, and we have H 0 : µ 1 = µ 2. And for the alternate hypothesis, the sample means are unequal, and we have H a : µ 1 ≠ µ 2.
• The observed population parameters and variables are the same across the samples, for a null hypothesis, but in an alternate hypothesis, there is a significant difference between the observed parameters and variables across the samples.

☛ Related Topics

The following topics help in a better understanding of the null hypothesis.

• Probability and Statistics
• Basic Statistics Formula
• Sample Space

Examples on Null Hypothesis

Example 1: A medical experiment and trial is conducted to check if a particular drug can serve as the vaccine for Covid-19, and can prevent from occurrence of Corona. Write the null hypothesis and the alternate hypothesis for this situation.

The given situation refers to a possible new drug and its effectiveness of being a vaccine for Covid-19 or not. The null hypothesis (H o ) and alternate hypothesis (H a ) for this medical experiment is as follows.

• H 0 : The use of the new drug is not helpful for the prevention of Covid-19.
• H a : The use of the new drug serves as a vaccine and helps for the prevention of Covid-19.

Example 2: The teacher has prepared a set of important questions and informs the student that preparing these questions helps in scoring more than 60% marks in the board exams. Write the null hypothesis and the alternate hypothesis for this situation.

The given situation refers to the teacher who has claimed that her important questions helps to score more than 60% marks in the board exams. The null hypothesis(H o ) and alternate hypothesis(H a ) for this situation is as follows.

• H o : The important questions given by the teacher does not really help the students to get a score of more than 60% in the board exams.
• H a : The important questions given by the teacher is helpful for the students to score more than 60% marks in the board exams.

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Practice Questions on Null Hypothesis

Faqs on null hypothesis, what is null hypothesis in maths.

Null hypothesis is used in statistics and it states if there is any significant difference between the two samples. The acceptance of null hypothesis mean that there is no significant difference between the two samples. And the rejection of null hypothesis means that the two samples are different, and we need to accept the alternate hypothesis. The null hypothesis statement is represented as H 0 and the alternate hypothesis is represented as H a .

How Do You Test Null Hypothesis?

The null hypothesis is broadly tested using two methods. The null hypothesis can be tested using significance testing and hypothesis testing.Broadly the test for null hypothesis is performed across four stages. First the null hypothesis is identified, secondly the null hypothesis is defined. Next a suitable test is used to test the hypothesis, and finally either the null hypothesis or the alternate hypothesis is accepted.

How To Accept or Reject Null Hypothesis?

The null hypothesis is accepted or rejected based on the result of the hypothesis testing. The p value is found and the significance level is defined. If the p-value is lesser than the significance level α, then the null hypothesis is rejected. And if the p-value is greater than the significance level α, then the null hypothesis is accepted.

What Is the Difference Between Null Hypothesis And Alternate Hypothesis?

The null hypothesis states that there is no significant difference between the two samples, and the alternate hypothesis states that there is a significant difference between the two samples. The null hypothesis is referred using H o and the alternate hypothesis is referred using H a . As per null hypothesis the observed variables and parameters are the same across the samples, but as per alternate hypothesis there is a significant difference between the observed variables and parameters across the samples.

What Is Null Hypothesis Example?

A few quick examples of null hypothesis are as follows.

• The salary of a person is independent of his profession, is an example of null hypothesis. And the salary is dependent on the profession of a person, is an alternate hypothesis.
• The performance of the students in Maths from two different classes is a null hypothesis. And the performance of the students from each of the classes is different, is an example of alternate hypothesis.
• The nutrient content of mango and a mango milk shake is equal and it can be taken as a null hypothesis. The test to prove the different nutrient content of the two is referred to as alternate hypothesis.

Exploring the Null Hypothesis: Definition and Purpose

Updated: July 5, 2023 by Ken Feldman

Hypothesis testing is a branch of statistics in which, using data from a sample, an inference is made about a population parameter or a population probability distribution .

First, a hypothesis statement and assumption is made about the population parameter or probability distribution. This initial statement is called the Null Hypothesis and is denoted by H o. An alternative or alternate hypothesis (denoted Ha ), is then stated which will be the opposite of the Null Hypothesis.

The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o. If the H o is rejected, the statistical conclusion is that the alternative or alternate hypothesis Ha is true.

Overview: What is the Null Hypothesis (Ho)? 

Hypothesis testing applies to all forms of statistical inquiry. For example, it can be used to determine whether there are differences between population parameters or an understanding about slopes of regression lines or equality of probability distributions.

In all cases, the first thing you do is state the Null and Alternate Hypotheses. The word Null in the context of hypothesis testing means “nothing” or “zero.”  

As an example, if we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of: 

Ho: mu1 = mu2 or mu1- mu2 = 0  

In other words, there is no difference, or the difference is zero. Note that the notation is in the form of a population parameter, not a sample statistic. 

Since you are using sample data to make your inferences about the population, it’s possible you’ll make an error. In the case of the Null Hypothesis, we can make one of two errors.

•   Type 1 , or alpha error: An alpha error is when you mistakenly reject the Null and believe that something significant happened. In other words, you believe that the means of the two populations are different when they aren’t.
• Type 2, or beta error: A beta error is when you fail to reject the null when you should have.  In this case, you missed something significant and failed to take action. 

A classic example is when you get the results back from your doctor after taking a blood test. If the doctor says you have an infection when you really don’t, that is an alpha error. That is thinking that there is something significant going on when there isn’t. We also call that a false positive. The doctor rejected the null that “there was zero infection” and missed the call.

On the other hand, if the doctor told you that everything was OK when you really did have an infection, then he made a beta, or type 2, error. He failed to reject the Null Hypothesis when he should have. That is called a false negative.

The decision to reject or not to reject the Null Hypothesis is based on three numbers. 

• Alpha, which you get to choose. Alpha is the risk you are willing to assume of falsely rejecting the Null. The typical values for alpha are 1%, 5%, or 10%. Depending on the importance of the conclusion, you only want to falsely claim a difference when there is none, 1%, 5%, or 10% of the time.
• Beta, which is typically 20%. This means you’re willing to be wrong 20% of the time in failing to reject the null when you should have. 
• P-value, which is calculated from the data. The p-value is the actual risk you have in being wrong if you reject the null. You would like that to be low.  

Your decision as to what to do about the null is made by comparing the alpha value (your assumed risk) with the p-value (actual risk). If the actual risk is lower than your assumed risk, you can feel comfortable in rejecting the null and claiming something has happened. But, if the actual risk is higher than your assumed risk you will be taking a bigger risk than you want by rejecting the null.

RELATED: NULL VS. ALTERNATIVE HYPOTHESIS

3 benefits of the null hypothesis .

The stating and testing of the null hypothesis is the foundation of hypothesis testing. By doing so, you set the parameters for your statistical inference.

1. Statistical assurance of determining differences between population parameters

Just looking at the mathematical difference between the means of two samples and making a decision is woefully inadequate. By statistically testing the null hypothesis, you will have more confidence in any inferences you want to make about populations based on your samples.

2. Statistically based estimation of the probability of a population distribution

Many statistical tests require assumptions of specific distributions. Many of these tests assume that the population follows the normal distribution . If it doesn’t, the test may be invalid.  

3. Assess the strength of your conclusions as to what to do with the null hypothesis

Hypothesis testing calculations will provide some relative strength to your decisions as to whether you reject or fail to reject the null hypothesis.

Why is the Null Hypothesis important to understand?

The interpretation of the statistics relative to the null hypothesis is what’s important.

1. Properly write the null hypothesis to properly capture what you are seeking to prove

The null is always written in the same format. That is, the lack of difference or some other condition. The alternative hypothesis can be written in three formats depending on what you want to prove. 

2. Frame your statement and select an appropriate alpha risk

You don’t want to place too big of a hurdle or burden on your decision-making relative to action on the null hypothesis by selecting an alpha value that is too high or too low.

3. There are decision errors when deciding on how to respond to the Null Hypothesis

Since your decision relative to rejecting or not rejecting the null is based on statistical calculations, it is important to understand how that decision works. 

An industry example of using the Null Hypothesis 

The new director of marketing just completed the rollout of a new marketing campaign targeting the Hispanic market. Early indications showed that the campaign was successful in increasing sales in the Hispanic market. 

He came to that conclusion by comparing a sample of sales prior to the campaign and current sales after implementation of the campaign. He was anxious to proudly tell his boss how successful the campaign was. But, he decided to first check with his Lean Six Sigma Black Belt to see whether she agreed with his conclusion.

The Black Belt first asked the director his tolerance for risk of being wrong by telling the boss the campaign was successful when in fact, it wasn’t. That was the alpha value. The Director picked 5% since he was new and didn’t want to make a false claim so early in his career. He also picked 20% as his beta value.  

When the Black Belt was done analyzing the data, she found out that the p-value was 15%.  That meant if the director told the VP the campaign worked, there was a 15% chance he would be wrong and that the campaign probably needed some revising. Since he was only willing to be wrong 5% of the time, the decision was to not reject the null since his 5% assumed risk was less than the 15% actual risk.

3 best practices when thinking about the Null Hypothesis 

Using hypothesis testing to help make better data-driven decisions requires that you properly address the Null Hypothesis. 

1. Always use the proper nomenclature when stating the Null Hypothesis 

The null will always be in the form of decisions regarding the population, not the sample. 

2. The Null Hypothesis will always be written as the absence of some parameter or process characteristic

The writing of the Alternate Hypothesis can vary, so be sure you understand exactly what condition you are testing against. 

3. Pick a reasonable alpha risk so you’re not always failing to reject the Null Hypothesis

Being too cautious will lead you to make beta errors, and you’ll never learn anything about your population data. 

Frequently Asked Questions (FAQ) about the Null Hypothesis

What form should the null hypothesis be written in.

The Null Hypothesis should always be in the form of no difference or zero and always refer to the state of the population, not the sample. 

What is an alpha error? 

An alpha error, or Type 1 error, is rejecting the Null Hypothesis and claiming a significant event has occurred when, in fact, that is not true and the Null should not have been rejected.

How do I use the alpha error and p-value to decide on what decision I should make about the Null Hypothesis? 

The most common way of answering this is, “If the p-value is low (less than the alpha), the Null should be rejected. If the p-value is high (greater than the alpha) then the Null should not be rejected.”

Becoming familiar with the Null Hypothesis (Ho)

The proper writing of the Null Hypothesis is the basis for applying hypothesis testing to help you make better data-driven decisions. The format of the Null will always be in the form of zero, or the non-existence of some condition. It will always refer to a population parameter and not the sample you use to do your hypothesis testing calculations.

Be aware of the two types of errors you can make when deciding on what to do with the Null. Select reasonable risks values for your alpha and beta risks. By comparing your alpha risk with the calculated risk computed from the data, you will have sufficient information to make a wise decision as to whether you should reject the Null Hypothesis or not.

About the Author

Ken Feldman

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Null hypothesis

Null hypothesis n., plural: null hypotheses [nʌl haɪˈpɒθɪsɪs] Definition: a hypothesis that is valid or presumed true until invalidated by a statistical test

Table of Contents

Null Hypothesis Definition

Null hypothesis is defined as “the commonly accepted fact (such as the sky is blue) and researcher aim to reject or nullify this fact”.

More formally, we can define a null hypothesis as “a statistical theory suggesting that no statistical relationship exists between given observed variables” .

In biology , the null hypothesis is used to nullify or reject a common belief. The researcher carries out the research which is aimed at rejecting the commonly accepted belief.

What Is a Null Hypothesis?

A hypothesis is defined as a theory or an assumption that is based on inadequate evidence. It needs and requires more experiments and testing for confirmation. There are two possibilities that by doing more experiments and testing, a hypothesis can be false or true. It means it can either prove wrong or true (Blackwelder, 1982).

For example, Susie assumes that mineral water helps in the better growth and nourishment of plants over distilled water. To prove this hypothesis, she performs this experiment for almost a month. She watered some plants with mineral water and some with distilled water.

In a hypothesis when there are no statistically significant relationships among the two variables, the hypothesis is said to be a null hypothesis. The investigator is trying to disprove such a hypothesis. In the above example of plants, the null hypothesis is:

There are no statistical relationships among the forms of water that are given to plants for growth and nourishment.

Usually, an investigator tries to prove the null hypothesis wrong and tries to explain a relation and association between the two variables.

An opposite and reverse of the null hypothesis are known as the alternate hypothesis . In the example of plants the alternate hypothesis is:

There are statistical relationships among the forms of water that are given to plants for growth and nourishment.

The example below shows the difference between null vs alternative hypotheses:

Alternate Hypothesis: The world is round Null Hypothesis: The world is not round.

Copernicus and many other scientists try to prove the null hypothesis wrong and false. By their experiments and testing, they make people believe that alternate hypotheses are correct and true. If they do not prove the null hypothesis experimentally wrong then people will not believe them and never consider the alternative hypothesis true and correct.

The alternative and null hypothesis for Susie’s assumption is:

• Null Hypothesis: If one plant is watered with distilled water and the other with mineral water, then there is no difference in the growth and nourishment of these two plants.
• Alternative Hypothesis:  If one plant is watered with distilled water and the other with mineral water, then the plant with mineral water shows better growth and nourishment.

The null hypothesis suggests that there is no significant or statistical relationship. The relation can either be in a single set of variables or among two sets of variables.

Most people consider the null hypothesis true and correct. Scientists work and perform different experiments and do a variety of research so that they can prove the null hypothesis wrong or nullify it. For this purpose, they design an alternate hypothesis that they think is correct or true. The null hypothesis symbol is H 0 (it is read as H null or H zero ).

Why is it named the “Null”?

The name null is given to this hypothesis to clarify and explain that the scientists are working to prove it false i.e. to nullify the hypothesis. Sometimes it confuses the readers; they might misunderstand it and think that statement has nothing. It is blank but, actually, it is not. It is more appropriate and suitable to call it a nullifiable hypothesis instead of the null hypothesis.

Why do we need to assess it? Why not just verify an alternate one?

In science, the scientific method is used. It involves a series of different steps. Scientists perform these steps so that a hypothesis can be proved false or true. Scientists do this to confirm that there will be any limitation or inadequacy in the new hypothesis. Experiments are done by considering both alternative and null hypotheses, which makes the research safe. It gives a negative as well as a bad impact on research if a null hypothesis is not included or a part of the study. It seems like you are not taking your research seriously and not concerned about it and just want to impose your results as correct and true if the null hypothesis is not a part of the study.

Development of the Null

In statistics, firstly it is necessary to design alternate and null hypotheses from the given problem. Splitting the problem into small steps makes the pathway towards the solution easier and less challenging. how to write a null hypothesis?

Writing a null hypothesis consists of two steps:

• Firstly, initiate by asking a question.
• Secondly, restate the question in such a way that it seems there are no relationships among the variables.

In other words, assume in such a way that the treatment does not have any effect.

The usual recovery duration after knee surgery is considered almost 8 weeks.

A researcher thinks that the recovery period may get elongated if patients go to a physiotherapist for rehabilitation twice per week, instead of thrice per week, i.e. recovery duration reduces if the patient goes three times for rehabilitation instead of two times.

Step 1: Look for the problem in the hypothesis. The hypothesis either be a word or can be a statement. In the above example the hypothesis is:

“The expected recovery period in knee rehabilitation is more than 8 weeks”

Step 2: Make a mathematical statement from the hypothesis. Averages can also be represented as μ, thus the null hypothesis formula will be.

In the above equation, the hypothesis is equivalent to H1, the average is denoted by μ and > that the average is greater than eight.

Step 3: Explain what will come up if the hypothesis does not come right i.e., the rehabilitation period may not proceed more than 08 weeks.

There are two options: either the recovery will be less than or equal to 8 weeks.

H 0 : μ ≤ 8

In the above equation, the null hypothesis is equivalent to H 0 , the average is denoted by μ and ≤ represents that the average is less than or equal to eight.

What will happen if the scientist does not have any knowledge about the outcome?

Problem: An investigator investigates the post-operative impact and influence of radical exercise on patients who have operative procedures of the knee. The chances are either the exercise will improve the recovery or will make it worse. The usual time for recovery is 8 weeks.

Step 1: Make a null hypothesis i.e. the exercise does not show any effect and the recovery time remains almost 8 weeks.

H 0 : μ = 8

In the above equation, the null hypothesis is equivalent to H 0 , the average is denoted by μ, and the equal sign (=) shows that the average is equal to eight.

Step 2: Make the alternate hypothesis which is the reverse of the null hypothesis. Particularly what will happen if treatment (exercise) makes an impact?

In the above equation, the alternate hypothesis is equivalent to H1, the average is denoted by μ and not equal sign (≠) represents that the average is not equal to eight.

Significance Tests

To get a reasonable and probable clarification of statistics (data), a significance test is performed. The null hypothesis does not have data. It is a piece of information or statement which contains numerical figures about the population. The data can be in different forms like in means or proportions. It can either be the difference of proportions and means or any odd ratio.

The following table will explain the symbols:

P-value is the chief statistical final result of the significance test of the null hypothesis.

• P-value = Pr(data or data more extreme | H 0 true)
• | = “given”
• Pr = probability
• H 0 = the null hypothesis

The first stage of Null Hypothesis Significance Testing (NHST) is to form an alternate and null hypothesis. By this, the research question can be briefly explained.

Null Hypothesis = no effect of treatment, no difference, no association Alternative Hypothesis = effective treatment, difference, association

When to reject the null hypothesis?

Researchers will reject the null hypothesis if it is proven wrong after experimentation. Researchers accept null hypothesis to be true and correct until it is proven wrong or false. On the other hand, the researchers try to strengthen the alternate hypothesis. The binomial test is performed on a sample and after that, a series of tests were performed (Frick, 1995).

Step 1: Evaluate and read the research question carefully and consciously and make a null hypothesis. Verify the sample that supports the binomial proportion. If there is no difference then find out the value of the binomial parameter.

Show the null hypothesis as:

H 0 :p= the value of p if H 0 is true

To find out how much it varies from the proposed data and the value of the null hypothesis, calculate the sample proportion.

Step 2: In test statistics, find the binomial test that comes under the null hypothesis. The test must be based on precise and thorough probabilities. Also make a list of pmf that apply, when the null hypothesis proves true and correct.

When H 0 is true, X~b(n, p)

N = size of the sample

P = assume value if H 0 proves true.

Step 3: Find out the value of P. P-value is the probability of data that is under observation.

Rise or increase in the P value = Pr(X ≥ x)

X = observed number of successes

P value = Pr(X ≤ x).

Step 4: Demonstrate the findings or outcomes in a descriptive detailed way.

• Sample proportion
• The direction of difference (either increases or decreases)

Perceived Problems With the Null Hypothesis

Variable or model selection and less information in some cases are the chief important issues that affect the testing of the null hypothesis. Statistical tests of the null hypothesis are reasonably not strong. There is randomization about significance. (Gill, 1999) The main issue with the testing of the null hypothesis is that they all are wrong or false on a ground basis.

There is another problem with the a-level . This is an ignored but also a well-known problem. The value of a-level is without a theoretical basis and thus there is randomization in conventional values, most commonly 0.q, 0.5, or 0.01. If a fixed value of a is used, it will result in the formation of two categories (significant and non-significant) The issue of a randomized rejection or non-rejection is also present when there is a practical matter which is the strong point of the evidence related to a scientific matter.

The P-value has the foremost importance in the testing of null hypothesis but as an inferential tool and for interpretation, it has a problem. The P-value is the probability of getting a test statistic at least as extreme as the observed one.

The main point about the definition is: Observed results are not based on a-value

Moreover, the evidence against the null hypothesis was overstated due to unobserved results. A-value has importance more than just being a statement. It is a precise statement about the evidence from the observed results or data. Similarly, researchers found that P-values are objectionable. They do not prefer null hypotheses in testing. It is also clear that the P-value is strictly dependent on the null hypothesis. It is computer-based statistics. In some precise experiments, the null hypothesis statistics and actual sampling distribution are closely related but this does not become possible in observational studies.

Some researchers pointed out that the P-value is depending on the sample size. If the true and exact difference is small, a null hypothesis even of a large sample may get rejected. This shows the difference between biological importance and statistical significance. (Killeen, 2005)

Another issue is the fix a-level, i.e., 0.1. On the basis, if a-level a null hypothesis of a large sample may get accepted or rejected. If the size of simple is infinity and the null hypothesis is proved true there are still chances of Type I error. That is the reason this approach or method is not considered consistent and reliable. There is also another problem that the exact information about the precision and size of the estimated effect cannot be known. The only solution is to state the size of the effect and its precision.

Null Hypothesis Examples

Here are some examples:

Example 1: Hypotheses with One Sample of One Categorical Variable

Among all the population of humans, almost 10% of people prefer to do their task with their left hand i.e. left-handed. Let suppose, a researcher in the Penn States says that the population of students at the College of Arts and Architecture is mostly left-handed as compared to the general population of humans in general public society. In this case, there is only a sample and there is a comparison among the known population values to the population proportion of sample value.

• Research Question: Do artists more expected to be left-handed as compared to the common population persons in society?
• Response Variable: Sorting the student into two categories. One category has left-handed persons and the other category have right-handed persons.
• Form Null Hypothesis: Arts and Architecture college students are no more predicted to be lefty as compared to the common population persons in society (Lefty students of Arts and Architecture college population is 10% or p= 0.10)

Example 2: Hypotheses with One Sample of One Measurement Variable

A generic brand of antihistamine Diphenhydramine making medicine in the form of a capsule, having a 50mg dose. The maker of the medicines is concerned that the machine has come out of calibration and is not making more capsules with the suitable and appropriate dose.

• Research Question: Does the statistical data recommended about the mean and average dosage of the population differ from 50mg?
• Response Variable: Chemical assay used to find the appropriate dosage of the active ingredient.
• Null Hypothesis: Usually, the 50mg dosage of capsules of this trade name (population average and means dosage =50 mg).

Example 3: Hypotheses with Two Samples of One Categorical Variable

Several people choose vegetarian meals on a daily basis. Typically, the researcher thought that females like vegetarian meals more than males.

• Research Question: Does the data recommend that females (women) prefer vegetarian meals more than males (men) regularly?
• Response Variable: Cataloguing the persons into vegetarian and non-vegetarian categories. Grouping Variable: Gender
• Null Hypothesis: Gender is not linked to those who like vegetarian meals. (Population percent of women who eat vegetarian meals regularly = population percent of men who eat vegetarian meals regularly or p women = p men).

Example 4: Hypotheses with Two Samples of One Measurement Variable

Nowadays obesity and being overweight is one of the major and dangerous health issues. Research is performed to confirm that a low carbohydrates diet leads to faster weight loss than a low-fat diet.

• Research Question: Does the given data recommend that usually, a low-carbohydrate diet helps in losing weight faster as compared to a low-fat diet?
• Response Variable: Weight loss (pounds)
• Explanatory Variable: Form of diet either low carbohydrate or low fat
• Null Hypothesis: There is no significant difference when comparing the mean loss of weight of people using a low carbohydrate diet to people using a diet having low fat. (population means loss of weight on a low carbohydrate diet = population means loss of weight on a diet containing low fat).

Example 5: Hypotheses about the relationship between Two Categorical Variables

A case-control study was performed. The study contains nonsmokers, stroke patients, and controls. The subjects are of the same occupation and age and the question was asked if someone at their home or close surrounding smokes?

• Research Question: Did second-hand smoke enhance the chances of stroke?
• Variables: There are 02 diverse categories of variables. (Controls and stroke patients) (whether the smoker lives in the same house). The chances of having a stroke will be increased if a person is living with a smoker.
• Null Hypothesis: There is no significant relationship between a passive smoker and stroke or brain attack. (odds ratio between stroke and the passive smoker is equal to 1).

Example 6: Hypotheses about the relationship between Two Measurement Variables

A financial expert observes that there is somehow a positive and effective relationship between the variation in stock rate price and the quantity of stock bought by non-management employees

• Response variable- Regular alteration in price
• Explanatory Variable- Stock bought by non-management employees
• Null Hypothesis: The association and relationship between the regular stock price alteration ($) and the daily stock-buying by non-management employees ($) = 0.

Example 7: Hypotheses about comparing the relationship between Two Measurement Variables in Two Samples

• Research Question: Is the relation between the bill paid in a restaurant and the tip given to the waiter, is linear? Is this relation different for dining and family restaurants?
• Explanatory Variable- total bill amount
• Response Variable- the amount of tip
• Null Hypothesis: The relationship and association between the total bill quantity at a family or dining restaurant and the tip, is the same.

Try to answer the quiz below to check what you have learned so far about the null hypothesis.

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• Blackwelder, W. C. (1982). “Proving the null hypothesis” in clinical trials. Controlled Clinical Trials , 3(4), 345–353.
• Frick, R. W. (1995). Accepting the null hypothesis. Memory & Cognition, 23(1), 132–138.
• Gill, J. (1999). The insignificance of null hypothesis significance testing. Political Research Quarterly , 52(3), 647–674.
• Killeen, P. R. (2005). An alternative to null-hypothesis significance tests. Psychological Science, 16(5), 345–353.

©BiologyOnline.com. Content provided and moderated by Biology Online Editors.

Last updated on June 16th, 2022

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Null Hypothesis

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Null Hypothesis , often denoted as H 0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring. The null is t he truth or falsity of an idea in analysis.

In this article, we will discuss the null hypothesis in detail, along with some solved examples and questions on the null hypothesis.

Table of Content

What is Null Hypothesis?

Null hypothesis symbol, formula of null hypothesis, types of null hypothesis, null hypothesis examples, principle of null hypothesis, how do you find null hypothesis, null hypothesis in statistics, null hypothesis and alternative hypothesis, null hypothesis and alternative hypothesis examples, null hypothesis – practice problems.

Null Hypothesis in statistical analysis suggests the absence of statistical significance within a specific set of observed data. Hypothesis testing, using sample data, evaluates the validity of this hypothesis. Commonly denoted as H 0 or simply “null,” it plays an important role in quantitative analysis, examining theories related to markets, investment strategies, or economies to determine their validity.

Null Hypothesis Meaning

Null Hypothesis represents a default position, often suggesting no effect or difference, against which researchers compare their experimental results. The Null Hypothesis, often denoted as H 0 asserts a default assumption in statistical analysis. It posits no significant difference or effect, serving as a baseline for comparison in hypothesis testing.

The null Hypothesis is represented as H 0 , the Null Hypothesis symbolizes the absence of a measurable effect or difference in the variables under examination.

Certainly, a simple example would be asserting that the mean score of a group is equal to a specified value like stating that the average IQ of a population is 100.

The Null Hypothesis is typically formulated as a statement of equality or absence of a specific parameter in the population being studied. It provides a clear and testable prediction for comparison with the alternative hypothesis. The formulation of the Null Hypothesis typically follows a concise structure, stating the equality or absence of a specific parameter in the population.

Mean Comparison (Two-sample t-test)

H 0 : μ 1 = μ 2

This asserts that there is no significant difference between the means of two populations or groups.

Proportion Comparison

H 0 : p 1 − p 2 = 0

This suggests no significant difference in proportions between two populations or conditions.

Equality in Variance (F-test in ANOVA)

H 0 : σ 1 = σ 2

This states that there’s no significant difference in variances between groups or populations.

Independence (Chi-square Test of Independence):

H 0 : Variables are independent

This asserts that there’s no association or relationship between categorical variables.

Null Hypotheses vary including simple and composite forms, each tailored to the complexity of the research question. Understanding these types is pivotal for effective hypothesis testing.

Equality Null Hypothesis (Simple Null Hypothesis)

The Equality Null Hypothesis, also known as the Simple Null Hypothesis, is a fundamental concept in statistical hypothesis testing that assumes no difference, effect or relationship between groups, conditions or populations being compared.

Non-Inferiority Null Hypothesis

In some studies, the focus might be on demonstrating that a new treatment or method is not significantly worse than the standard or existing one.

Superiority Null Hypothesis

The concept of a superiority null hypothesis comes into play when a study aims to demonstrate that a new treatment, method, or intervention is significantly better than an existing or standard one.

Independence Null Hypothesis

In certain statistical tests, such as chi-square tests for independence, the null hypothesis assumes no association or independence between categorical variables.

Homogeneity Null Hypothesis

In tests like ANOVA (Analysis of Variance), the null hypothesis suggests that there’s no difference in population means across different groups.

• Medicine: Null Hypothesis: “No significant difference exists in blood pressure levels between patients given the experimental drug versus those given a placebo.”
• Education: Null Hypothesis: “There’s no significant variation in test scores between students using a new teaching method and those using traditional teaching.”
• Economics: Null Hypothesis: “There’s no significant change in consumer spending pre- and post-implementation of a new taxation policy.”
• Environmental Science: Null Hypothesis: “There’s no substantial difference in pollution levels before and after a water treatment plant’s establishment.”

The principle of the null hypothesis is a fundamental concept in statistical hypothesis testing. It involves making an assumption about the population parameter or the absence of an effect or relationship between variables.

In essence, the null hypothesis (H 0 ) proposes that there is no significant difference, effect, or relationship between variables. It serves as a starting point or a default assumption that there is no real change, no effect or no difference between groups or conditions.

Null Hypothesis Rejection

Rejecting the Null Hypothesis occurs when statistical evidence suggests a significant departure from the assumed baseline. It implies that there is enough evidence to support the alternative hypothesis, indicating a meaningful effect or difference. Null Hypothesis rejection occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

Identifying the Null Hypothesis involves defining the status quotient, asserting no effect and formulating a statement suitable for statistical analysis.

When is Null Hypothesis Rejected?

The Null Hypothesis is rejected when statistical tests indicate a significant departure from the expected outcome, leading to the consideration of alternative hypotheses. It occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

In statistical hypothesis testing, researchers begin by stating the null hypothesis, often based on theoretical considerations or previous research. The null hypothesis is then tested against an alternative hypothesis (Ha), which represents the researcher’s claim or the hypothesis they seek to support.

The process of hypothesis testing involves collecting sample data and using statistical methods to assess the likelihood of observing the data if the null hypothesis were true. This assessment is typically done by calculating a test statistic, which measures the difference between the observed data and what would be expected under the null hypothesis.

In the realm of hypothesis testing, the null hypothesis (H 0 ) and alternative hypothesis (H₁ or Ha) play critical roles. The null hypothesis generally assumes no difference, effect, or relationship between variables, suggesting that any observed change or effect is due to random chance. Its counterpart, the alternative hypothesis, asserts the presence of a significant difference, effect, or relationship between variables, challenging the null hypothesis. These hypotheses are formulated based on the research question and guide statistical analyses.

Difference Between Null Hypothesis and Alternative Hypothesis

The null hypothesis (H 0 ) serves as the baseline assumption in statistical testing, suggesting no significant effect, relationship, or difference within the data. It often proposes that any observed change or correlation is merely due to chance or random variation. Conversely, the alternative hypothesis (H 1 or Ha) contradicts the null hypothesis, positing the existence of a genuine effect, relationship or difference in the data. It represents the researcher’s intended focus, seeking to provide evidence against the null hypothesis and support for a specific outcome or theory. These hypotheses form the crux of hypothesis testing, guiding the assessment of data to draw conclusions about the population being studied.

Let’s envision a scenario where a researcher aims to examine the impact of a new medication on reducing blood pressure among patients. In this context:

Null Hypothesis (H 0 ): “The new medication does not produce a significant effect in reducing blood pressure levels among patients.”

Alternative Hypothesis (H 1 or Ha): “The new medication yields a significant effect in reducing blood pressure levels among patients.”

The null hypothesis implies that any observed alterations in blood pressure subsequent to the medication’s administration are a result of random fluctuations rather than a consequence of the medication itself. Conversely, the alternative hypothesis contends that the medication does indeed generate a meaningful alteration in blood pressure levels, distinct from what might naturally occur or by random chance.

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Example 1: A researcher claims that the average time students spend on homework is 2 hours per night.

Null Hypothesis (H 0 ): The average time students spend on homework is equal to 2 hours per night. Data: A random sample of 30 students has an average homework time of 1.8 hours with a standard deviation of 0.5 hours. Test Statistic and Decision: Using a t-test, if the calculated t-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: Based on the statistical analysis, we fail to reject the null hypothesis, suggesting that there is not enough evidence to dispute the claim of the average homework time being 2 hours per night.

Example 2: A company asserts that the error rate in its production process is less than 1%.

Null Hypothesis (H 0 ): The error rate in the production process is 1% or higher. Data: A sample of 500 products shows an error rate of 0.8%. Test Statistic and Decision: Using a z-test, if the calculated z-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: The statistical analysis supports rejecting the null hypothesis, indicating that there is enough evidence to dispute the company’s claim of an error rate of 1% or higher.

Q1. A researcher claims that the average time spent by students on homework is less than 2 hours per day. Formulate the null hypothesis for this claim?

Q2. A manufacturing company states that their new machine produces widgets with a defect rate of less than 5%. Write the null hypothesis to test this claim?

Q3. An educational institute believes that their online course completion rate is at least 60%. Develop the null hypothesis to validate this assertion?

Q4. A restaurant claims that the waiting time for customers during peak hours is not more than 15 minutes. Formulate the null hypothesis for this claim?

Q5. A study suggests that the mean weight loss after following a specific diet plan for a month is more than 8 pounds. Construct the null hypothesis to evaluate this statement?

Summary – Null Hypothesis and Alternative Hypothesis

The null hypothesis (H 0 ) and alternative hypothesis (H a ) are fundamental concepts in statistical hypothesis testing. The null hypothesis represents the default assumption, stating that there is no significant effect, difference, or relationship between variables. It serves as the baseline against which the alternative hypothesis is tested. In contrast, the alternative hypothesis represents the researcher’s hypothesis or the claim to be tested, suggesting that there is a significant effect, difference, or relationship between variables. The relationship between the null and alternative hypotheses is such that they are complementary, and statistical tests are conducted to determine whether the evidence from the data is strong enough to reject the null hypothesis in favor of the alternative hypothesis. This decision is based on the strength of the evidence and the chosen level of significance. Ultimately, the choice between the null and alternative hypotheses depends on the specific research question and the direction of the effect being investigated.

FAQs on Null Hypothesis

What does null hypothesis stands for.

The null hypothesis, denoted as H 0 ​, is a fundamental concept in statistics used for hypothesis testing. It represents the statement that there is no effect or no difference, and it is the hypothesis that the researcher typically aims to provide evidence against.

How to Form a Null Hypothesis?

A null hypothesis is formed based on the assumption that there is no significant difference or effect between the groups being compared or no association between variables being tested. It often involves stating that there is no relationship, no change, or no effect in the population being studied.

When Do we reject the Null Hypothesis?

In statistical hypothesis testing, if the p-value (the probability of obtaining the observed results) is lower than the chosen significance level (commonly 0.05), we reject the null hypothesis. This suggests that the data provides enough evidence to refute the assumption made in the null hypothesis.

What is a Null Hypothesis in Research?

In research, the null hypothesis represents the default assumption or position that there is no significant difference or effect. Researchers often try to test this hypothesis by collecting data and performing statistical analyses to see if the observed results contradict the assumption.

What Are Alternative and Null Hypotheses?

The null hypothesis (H0) is the default assumption that there is no significant difference or effect. The alternative hypothesis (H1 or Ha) is the opposite, suggesting there is a significant difference, effect or relationship.

What Does it Mean to Reject the Null Hypothesis?

Rejecting the null hypothesis implies that there is enough evidence in the data to support the alternative hypothesis. In simpler terms, it suggests that there might be a significant difference, effect or relationship between the groups or variables being studied.

How to Find Null Hypothesis?

Formulating a null hypothesis often involves considering the research question and assuming that no difference or effect exists. It should be a statement that can be tested through data collection and statistical analysis, typically stating no relationship or no change between variables or groups.

How is Null Hypothesis denoted?

The null hypothesis is commonly symbolized as H 0 in statistical notation.

What is the Purpose of the Null hypothesis in Statistical Analysis?

The null hypothesis serves as a starting point for hypothesis testing, enabling researchers to assess if there’s enough evidence to reject it in favor of an alternative hypothesis.

What happens if we Reject the Null hypothesis?

Rejecting the null hypothesis implies that there is sufficient evidence to support an alternative hypothesis, suggesting a significant effect or relationship between variables.

What are Test for Null Hypothesis?

Various statistical tests, such as t-tests or chi-square tests, are employed to evaluate the validity of the Null Hypothesis in different scenarios.

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What Is a Null Hypothesis?

The alternative hypothesis.

• Additional Examples
• Null Hypothesis and Investments

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Null Hypothesis: What Is It, and How Is It Used in Investing?

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

A null hypothesis is a type of statistical hypothesis that proposes that no statistical significance exists in a set of given observations. Hypothesis testing is used to assess the credibility of a hypothesis by using sample data. Sometimes referred to simply as the “null,” it is represented as H 0 .

The null hypothesis, also known as the conjecture, is used in quantitative analysis to test theories about markets, investing strategies, or economies to decide if an idea is true or false.

Key Takeaways

• A null hypothesis is a type of conjecture in statistics that proposes that there is no difference between certain characteristics of a population or data-generating process.
• The alternative hypothesis proposes that there is a difference.
• Hypothesis testing provides a method to reject a null hypothesis within a certain confidence level.
• If you can reject the null hypothesis, it provides support for the alternative hypothesis.
• Null hypothesis testing is the basis of the principle of falsification in science.

Alex Dos Diaz / Investopedia

Null Hypothesis Example

For example, a gambler may be interested in whether a game of chance is fair. If it is fair, then the expected earnings per play come to zero for both players. If the game is not fair, then the expected earnings are positive for one player and negative for the other.

To test whether the game is fair, the gambler collects earnings data from many repetitions of the game, calculates the average earnings from these data, then tests the null hypothesis that the expected earnings are not different from zero.

If the average earnings from the sample data are sufficiently far from zero, then the gambler will reject the null hypothesis and conclude the alternative hypothesis—namely, that the expected earnings per play are different from zero. If the average earnings from the sample data are near zero, then the gambler will not reject the null hypothesis, concluding instead that the difference between the average from the data and zero is explainable by chance alone.

The null hypothesis assumes that any kind of difference between the chosen characteristics that you see in a set of data is due to chance. For example, if the expected earnings for the gambling game are truly equal to zero, then any difference between the average earnings in the data and zero is due to chance.

Analysts look to reject   the null hypothesis because doing so is a strong conclusion. This requires strong evidence in the form of an observed difference that is too large to be explained solely by chance. Failing to reject the null hypothesis—that the results are explainable by chance alone—is a weak conclusion because it allows that factors other than chance may be at work, but may not be strong enough for the statistical test to detect them.

A null hypothesis can only be rejected, not proven.

An important point to note is that we are testing the null hypothesis because there is an element of doubt about its validity. Whatever information that is against the stated null hypothesis is captured in the alternative (alternate) hypothesis (H1).

For the examples below, the alternative hypothesis would be:

• Students score an average that is  not  equal to seven.
• The mean annual return of a mutual fund is  not  equal to 8% per year.

In other words, the alternative hypothesis is a direct contradiction of the null hypothesis.

More Null Hypothesis Examples

Here is a simple example: A school principal claims that students in her school score an average of seven out of 10 in exams. The null hypothesis is that the population mean is 7.0. To test this null hypothesis, we record marks of, say, 30 students ( sample ) from the entire student population of the school (say, 300) and calculate the mean of that sample.

We can then compare the (calculated) sample mean to the (hypothesized) population mean of 7.0 and attempt to reject the null hypothesis. (The null hypothesis here—that the population mean is 7.0—cannot be proved using the sample data. It can only be rejected.)

Take another example: The annual return of a particular  mutual fund  is claimed to be 8%. Assume that a mutual fund has been in existence for 20 years. The null hypothesis is that the mean return is 8% for the mutual fund. We take a random sample of annual returns of the mutual fund for, say, five years (sample) and calculate the sample mean. We then compare the (calculated) sample mean to the (claimed) population mean (8%) to test the null hypothesis.

For the above examples, null hypotheses are:

• Example A : Students in the school score an average of seven out of 10 in exams.
• Example B : The mean annual return of the mutual fund is 8% per year.

For the purposes of determining whether to reject the null hypothesis, the null hypothesis (abbreviated H 0 ) is assumed, for the sake of argument, to be true. Then the likely range of possible values of the calculated statistic (e.g., the average score on 30 students’ tests) is determined under this presumption (e.g., the range of plausible averages might range from 6.2 to 7.8 if the population mean is 7.0). Then, if the sample average is outside of this range, the null hypothesis is rejected. Otherwise, the difference is said to be “explainable by chance alone,” being within the range that is determined by chance alone.

How Null Hypothesis Testing Is Used in Investments

As an example related to financial markets, assume Alice sees that her investment strategy produces higher average returns than simply buying and holding a stock . The null hypothesis states that there is no difference between the two average returns, and Alice is inclined to believe this until she can conclude contradictory results.

Refuting the null hypothesis would require showing statistical significance, which can be found by a variety of tests. The alternative hypothesis would state that the investment strategy has a higher average return than a traditional buy-and-hold strategy.

One tool that can determine the statistical significance of the results is the p-value. A p-value represents the probability that a difference as large or larger than the observed difference between the two average returns could occur solely by chance.

A p-value that is less than or equal to 0.05 often indicates whether there is evidence against the null hypothesis. If Alice conducts one of these tests, such as a test using the normal model, resulting in a significant difference between her returns and the buy-and-hold returns (the p-value is less than or equal to 0.05), she can then reject the null hypothesis and conclude the alternative hypothesis.

How Is the Null Hypothesis Identified?

The analyst or researcher establishes a null hypothesis based on the research question or problem that they are trying to answer. Depending on the question, the null may be identified differently. For example, if the question is simply whether an effect exists (e.g., does X influence Y?), the null hypothesis could be H 0 : X = 0. If the question is instead, is X the same as Y, the H0 would be X = Y. If it is that the effect of X on Y is positive, H0 would be X > 0. If the resulting analysis shows an effect that is statistically significantly different from zero, the null can be rejected.

How Is Null Hypothesis Used in Finance?

In finance , a null hypothesis is used in quantitative analysis. A null hypothesis tests the premise of an investing strategy, the markets, or an economy to determine if it is true or false.

For instance, an analyst may want to see if two stocks, ABC and XYZ, are closely correlated. The null hypothesis would be ABC ≠ XYZ.

How Are Statistical Hypotheses Tested?

Statistical hypotheses are tested by a four-step process . The first step is for the analyst to state the two hypotheses so that only one can be right. The next step is to formulate an analysis plan, which outlines how the data will be evaluated. The third step is to carry out the plan and physically analyze the sample data. The fourth and final step is to analyze the results and either reject the null hypothesis or claim that the observed differences are explainable by chance alone.

What Is an Alternative Hypothesis?

An alternative hypothesis is a direct contradiction of a null hypothesis. This means that if one of the two hypotheses is true, the other is false.

A null hypothesis is a type of statistical hypothesis. It proposes that no statistical significance exists in a set of given observations.

Also known as the conjecture, the null hypothesis is used in quantitative analysis to test theories about economies, investing strategies, or markets to decide if an idea is true or false. Hypothesis testing assesses the credibility of a hypothesis by using sample data. It is represented as H0 and is sometimes simply known as the “null.”

Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Page 4.

Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Pages 4–7.

Sage Publishing. “ Chapter 8: Introduction to Hypothesis Testing ,” Page 7.

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What does "Null Hypothesis" mean?

Definition of Null Hypothesis in the context of A/B testing (online controlled experiments).

What is a Null Hypothesis?

The null hypothesis in the colloquial sense is a claim that we want to put to the test . It is usually the default, inherited state of matters, or a state of matters that would be preferred if no further information is available (we do not test). In other words, it should correspond to the position of a critic, skeptical to our proposed alternative or new treatment (the alternative hypothesis ). In the strict statistical sense of the term it is defined more strictly as a statistical model (see hypothesis ).

The null hypothesis is usually denoted by H 0 .

There are different kinds of null hypotheses, depending on the inference we want to make in the case at hand but in all cases the null and alternative should exhaust the entire parameter space (Θ 0 ∪ Θ 1 = Θ, Θ 0 ∩ Θ 1 = ∅). Most often in A/B tests we set up a superiority alternative such that the null hypothesis becomes H 0 : μ ≤ 0 or H 0 : μ ≤ m where m is some positive discrepancy. However, sometimes the alternative is that of non-inferiority so the null becomes H 0 ≤ -m where m is the magnitude of the non-inferiority margin .

The null hypothesis should not be confused with the nil hypothesis (no difference hypothesis) although the two can coincide in the case of particular two-sided tests. The null hypothesis can be defined as any point or range which corresponds to a claim that needs to be tested, e.g. μ ≤ 2.

From a practical standpoint the null hypothesis is what gives the p-value a meaningful interpretation since the probability expressed by it is calculated under the assumption that the null is true. Upon observing a p-value below our chosen significance threshold we have sufficient evidence to reject the null hypothesis using the modus tollens logic or the argument from coincidence . We effectively shift the burden of proof to the one who would wand to make a claim corresponding to the null hypothesis.

The null can be redefined after a test is completed in order to check how warranted different conclusions are. For example, if you observed a given mean value x and the result is statistically significant for the original H 0 : μ 0 : μ ≤ x then you know immediately that you have poor evidence to reject H1 0 since the probability of observing this value assuming H1 0 is true is exactly 0.5, thus very likely.

A confidence interval can also be used to test a null hypothesis: every null hypothesis defined over any set of values not covered by the confidence interval can be rejected at significance level equal to 1 minus the confidence level , e.g. if a 95% confidence interval covers the values from 0.01 to +∞ then a null that covers the values from -∞ to 0.005 can be rejected at the 1-0.95 = 0.05 level.

Like this glossary entry? For an in-depth and comprehensive reading on A/B testing stats, check out the book "Statistical Methods in Online A/B Testing" by the author of this glossary, Georgi Georgiev.

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• What is Null Hypothesis

What is a Null Hypothesis?

In Maths, Statistics is a concept which deals with research and analysis of numerical data. We can consider hypothesis when we are not provided with accurate data sets . Basically, there are two types of hypotheses. They are the Null Hypothesis and Alternative Hypothesis. A hypothesis is a consideration or theory based on inadequate evidence that confers itself to advance testing and experimentation. With additional testing, a prediction can generally be demonstrated as true or false. In this article, we are going to learn what is null hypothesis in statistics? null hypothesis formula, symbol, the difference between the null hypothesis and alternative hypothesis and examples in detail.

Table of Contents:

• Difference Between Null and Alternative hypothesis

What is Null Hypothesis in Statistics?

A null hypothesis is a theory that assumes there is no statistical importance between the two variables in the hypothesis. It is the assumption that the researcher is seeking to expose. For example, there is no statistically meaningful relationship between the type of water fed to the plants and growth of the plants. A researcher is questioned by the null hypothesis and normally wants to deny it, to illustrate that there is a statistically vital relationship between the two variables in the hypothesis.

Learn in detail about Null Hypothesis here.

The null hypothesis is sometimes rejected too. If this assumption is rejected, it means that analysis could be unreasonable. Many researchers will ignore this hypothesis as it is slightly opposite to the alternative hypothesis . It is a better method to create a theory and test it. The aim of researchers is not to discard the hypothesis. But it is proof that a certain statistical model is always connected with the failure to decline the null hypothesis.

In statistics, the symbol of the null hypothesis is denoted by, H 0 , i.e., letter H with subscript ‘0’ (zero). It is referred to as H-null or H-zero or H-nought. Whereas, the alternative hypothesis represents the observations defined by the non-random condition. It is represented by H 1 or H a .

Also, read:

• Data Collection And Organization

Null hypothesis formula:

H 0 : p = p 0

p=mean of population 1

p 0 =mean of population 2

Alternative hypothesis formula is:

H a : p ≠ p 0 (p >p 0 )

Test static formula is:

Null and Alternative hypothesis

Alternative hypothesis defines there is a statistically meaningful relationship between two quantities or variables. Whereas null hypothesis states, there is no statistical correlation between the two variables.

The conclusions of null hypothesis are the outcome of possibility and the result of the alternative hypothesis is the outcome of real effect.

The mathematical formulation is an equal sign for null hypothesis, whereas for the alternative hypothesis, it is an inequality sign like greater than (>), less than (<), etc

Examples of Null Hypothesis

• There could be the possibility of getting deceased by typhoid but not 100%.
• There is no age limit to using mobile phones to access the internet.
• Having an apple a day does not ensure that we would not get a fever, but it helps to boost immunity to fight against the disease.

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Frequently Asked Questions on What is Null Hypothesis

What is null hypothesis in statistics.

The null hypothesis is one of the types of hypotheses in Statistics which proposes that there is no relationship between certain characteristics of a population.

Why is the null hypothesis important?

The null hypothesis is important because it helps us to conclude whether or not there is a relationship between two measured phenomena.

What is the difference between the null hypothesis and alternative hypothesis?

Alternative hypothesis describes that there is a statistically meaningful relationship between two quantities or variables. Whereas the null hypothesis states that there is no statistical correlation between the two variables.

What is the symbol used for the null hypothesis?

The symbol used for the null hypothesis is H 0 and it is pronounced as H-null or H-zero or H-nought.

What is the symbol used for the alternative hypothesis?

The symbol used for the alternative hypothesis is H a or H 1 .

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What is a Null Hypothesis?

Home › Economics › Macroeconomics › What is a Null Hypothesis?

Definition: A null hypothesis is a statistical assumption that suggests that two variables don’t affect each other in any way. It is a hypothesis that needs to be proven in order to be either accepted or rejected.

• What Does Null Hypothesis Mean?

This hypothesis starts by assuming, in a negative sense, that a relation between the variables being studied doesn’t exist. In order to prove or disprove the hypothesis a test must be made. The most common intention of the researcher is to disprove the null hypothesis, but the result might also be that the null hypothesis is confirmed. Test results will define the accuracy of the statement. Null hypothesis are used in combination with alternative hypothesis.

The alternative hypothesis disagrees with the null one and the researcher will focus on disprove the null hypothesis in order to approve the alternative hypothesis. It doesn’t mean the alternative hypothesis is correct, but by discarding the new one the research process can continue until one of the hypotheses is absolutely confirmed.

The Marketing Manager of a company called High Sound LLC is currently researching the relationship between two products that are sold together, a guitar and a guitar maintenance kit. The research project started when the owner of the company approached the Manager with a null hypothesis about the interaction between the sales of two different products.

The null hypothesis was: “There is no impact in the monthly number of guitar maintenance kits being sold if we sell more guitars”. After being addressed with this hypothesis the Marketing Manager formulated an alternative one that he would like to prove.

His alternative hypothesis was: “The number of guitar maintenance kits sold in a month increases when there’s an increase in the number of guitars sold in a month”. By testing the null hypothesis the Marketing Manager will be able to prove or disprove his own alternative hypothesis.

After conducting the test, the results showed that the null hypothesis was not correct; this meant that the alternative hypothesis became the accepted one.

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• Published: 19 June 2024

Amazon forest biogeography predicts resilience and vulnerability to drought

• Shuli Chen   ORCID: orcid.org/0000-0001-9852-8716 1 ,
• Scott C. Stark   ORCID: orcid.org/0000-0002-1579-1648 2 ,
• Antonio Donato Nobre 3 ,
• Luz Adriana Cuartas 4 ,
• Diogo de Jesus Amore 4 ,
• Natalia Restrepo-Coupe   ORCID: orcid.org/0000-0003-3921-1772 1 , 5 ,
• Marielle N. Smith   ORCID: orcid.org/0000-0003-2323-331X 2 , 6 ,
• Rutuja Chitra-Tarak 7 ,
• Hongseok Ko 1 ,
• Bruce W. Nelson   ORCID: orcid.org/0000-0002-0488-6895 8 &
• Scott R. Saleska 1 , 9  

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• Climate-change ecology
• Ecosystem ecology

Amazonia contains the most extensive tropical forests on Earth, but Amazon carbon sinks of atmospheric CO 2 are declining, as deforestation and climate-change-associated droughts 1 , 2 , 3 , 4 threaten to push these forests past a tipping point towards collapse 5 , 6 , 7 , 8 . Forests exhibit complex drought responses, indicating both resilience (photosynthetic greening) and vulnerability (browning and tree mortality), that are difficult to explain by climate variation alone 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 . Here we combine remotely sensed photosynthetic indices with ground-measured tree demography to identify mechanisms underlying drought resilience/vulnerability in different intact forest ecotopes 18 , 19 (defined by water-table depth, soil fertility and texture, and vegetation characteristics). In higher-fertility southern Amazonia, drought response was structured by water-table depth, with resilient greening in shallow-water-table forests (where greater water availability heightened response to excess sunlight), contrasting with vulnerability (browning and excess tree mortality) over deeper water tables. Notably, the resilience of shallow-water-table forest weakened as drought lengthened. By contrast, lower-fertility northern Amazonia, with slower-growing but hardier trees (or, alternatively, tall forests, with deep-rooted water access), supported more-drought-resilient forests independent of water-table depth. This functional biogeography of drought response provides a framework for conservation decisions and improved predictions of heterogeneous forest responses to future climate changes, warning that Amazonia’s most productive forests are also at greatest risk, and that longer/more frequent droughts are undermining multiple ecohydrological strategies and capacities for Amazon forest resilience.

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Divergent forest sensitivity to repeated extreme droughts

Basin-wide variation in tree hydraulic safety margins predicts the carbon balance of Amazon forests

Sensitivity of South American tropical forests to an extreme climate anomaly

Data availability.

All remote sensing data and products (vegetation/photosynthetic indices ( https://lpdaac.usgs.gov/products/mcd19a3v006/ , http://data.globalecology.unh.edu/data/GOSIF_v2 ), climate variables ( https://disc2.gesdisc.eosdis.nasa.gov/data/TRMM_L3/TRMM_3B43.7/ , https://goldsmr4.gesdisc.eosdis.nasa.gov/data/MERRA2_MONTHLY/M2TMNXRAD.5.12.4/ , https://airs.jpl.nasa.gov/data/get-data/standard-data/ ), land cover ( https://lpdaac.usgs.gov/products/mcd12q1v006/ , https://forobs.jrc.ec.europa.eu/TMF ), tree characteristics (canopy height, https://webmap.ornl.gov/ogc/dataset.jsp?dg_id=10023_1 ) and soil texture ( https://maps.isric.org/ )) are publicly available online. The ground-based demographic validation data are publicly available in refs. 2 , 26 . The ground-based hydraulic trait validation data are publicly available in ref. 50 . The HAND data are from ref. 25 , which derived them from the digital elevation model from the Shuttle Radar Topography Mission. The soil fertility data are available in ref. 43 .

Code availability

Code for reproducing the modelling analysis and figures is posted at Code Ocean ( https://codeocean.com/capsule/2432086/tree ).

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Acknowledgements

We thank J. Schietti for early discussions of the idea that remote sensing might be used to investigate the effect of water-table depth on forest drought response; T. R. Sousa for sharing and discussing plot-based forest demographic data (from along the BR-319 road) 26 ; G. Zuquim for sharing an early version of mapped basin-wide soil fertility data 43 ; H. ter Steege for sharing mapped basin-wide tree characteristics data 34 ; T. R. Sousa and J. Schietti for comments on an earlier version of the manuscript; L. Alves for advice on forest demography plots; R. Palacios for recommending the use of GAM models; M. N. Garcia for discussions about soil fertility; N. Boers for advice on the South American monsoon system; T. C. Taylor and V. Ivanov for discussions; J. Cronin and S. McMahon for detailed advice and comments; and S.C.’s doctoral dissertation committee members W. K. Smith, J. Hu and B. Enquist for constructive criticism and advice on the direction of this work. This work was supported by US National Aeronautics and Space Administration, fellowship 80NSSC19K1376 (S.C.); US National Science Foundation, DEB grant 1950080 (S.C.S. and M.N.S.); US National Science Foundation, DEB grant 2015832 (S.R.S.); US National Science Foundation, DEB grant 1754803 (S.R.S., N.R.-C.), US National Science Foundation, DEB grant 1754357 (S.C.S.); Brazil National Council for Scientific and Technological Development (CNPq) scholarships 371626/2022-6, 372734/2021-9,381711/2020-0 (D.d.J.A.); and US Department of Energy’s Next Generation Ecosystem Experiments-Tropics (R.C.-T.).

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Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ, USA

Shuli Chen, Natalia Restrepo-Coupe, Hongseok Ko & Scott R. Saleska

Department of Forestry, Michigan State University, East Lansing, MI, USA

Scott C. Stark & Marielle N. Smith

National Institute for Space Research (INPE), São José dos Campos, Brazil

Antonio Donato Nobre

National Center for Monitoring and Early Warning of Natural Disasters (CEMADEN), São José dos Campos, Brazil

Luz Adriana Cuartas & Diogo de Jesus Amore

Cupoazu LLC, Etobicoke, Ontario, Canada

Natalia Restrepo-Coupe

School of Environmental and Natural Sciences, College of Science and Engineering, Bangor University, Bangor, UK

Marielle N. Smith

Los Alamos National Laboratory, Earth and Environmental Sciences, Los Alamos, NM, USA

Rutuja Chitra-Tarak

Brazil’s National Institute for Amazon Research (INPA), Manaus, Brazil

Bruce W. Nelson

Department of Environmental Sciences, University of Arizona, Tucson, AZ, USA

Scott R. Saleska

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Contributions

S.C. and S.R.S. designed the analysis, based on early conception by A.D.N. and S.R.S., and on funded proposals to investigate ‘the other side of tropical forest drought’ led by S.C.S., M.N.S. and S.R.S. (from NSF) and by S.C. and S.R.S. (from NASA). A.D.N., L.A.C. and D.d.J.A. updated their HAND data product and interpreted it for this analysis. B.W.N. and N.R.-C. contributed remote-sensing expertise and analysis. R.C.-T. contributed statistical modelling expertise and analysis. H.K. contributed code, especially for the variogram analysis. S.C. organized the datasets (with assistance from N.R.-C.), conducted the analysis and wrote the initial draft. S.C., S.R.S. and S.C.S. revised the draft. All of the authors contributed to writing the final version.

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Extended data figures and tables

Extended data fig. 1 drought maps 2005, 2010 and 2015/16 droughts and gosif-based forest responses droughts..

(a)-(c): Maximum cumulative water deficit (MCWD) standardized anomalies. (relative to the long term mean MCWD across years, blue=positive, orange=negative) during drought for ( a ) 2005, ( b ) 2010, and ( c ) 2015 droughts. MCWD is calculated (see Methods, ‘Climate variables’) as the maximum water deficit reached for each hydrologic year (from May of the nominal year to the following April). The “drought region” is defined as pixels whose MCWD anomaly is more than one SD below the mean (light orange to red). (d)-(f): GOSIF-based forest response to droughts . GOSIF anomalies during drought, relative to the long term mean GOSIF (green=positive, orange=negative) in drought regions for the ( d ) 2005, ( e ) 2010 and ( f ) 2015 droughts, respectively. ( g ) EVI (left axis) and GOSIF (right axis) anomalies in the 2005 drought elliptical region (as depicted in Figs. 1a , 2a , and here in Extended Data Fig. 1d ) show consistent patterns versus HAND (bin averages ±95% CI, with N = 6,547 5-km pixels for both EVI and GOSIF); ( h ) GOSIF anomalies (bin averages points ±95% CI and solid regression line) vs. water-table depths (indexed by HAND) support hypothesis 1 (with negative slopes, consistent with EVI in Fig. 3a ) for the 2005 (green, slope = −0.016 ± 0.006 SE m −1 ), 2010 (purple, slope = −0.012 ± 0.003 SE m −1 ), and 2015 (blue, slope = −0.010 ± 0.003 SE m −1 ) droughts, paired with HAND distributions in each drought region (bottom graphs, right axis, with N = 34,980, 30,004, 43,475 5-km pixels for 2005, 2010, and 2015 droughts, respectively).

Extended Data Fig. 2 Ecotope factors of the Amazon basin.

( a ) Height Above Nearest Drainage (HAND), a proxy for water-table depth 25 ; ( b ) Soil fertility, as exchangeable base cation concentrations 43 ; ( c ) Average forest heights as acquired by lidar 45 ; ( d ) Soil sand content 44 ; ( e ) Proportion of trees belonging to the Fabaceae family 34 ; ( f ) MCWD variability (see the ‘Climate anomalies for drought definition and mapping’ section of  methods ), in terms of the standard deviation of the long-term MCWD timeseries. High variance in climate and low soil fertility in Guiana shield might contribute to the greatest proportion of trees belonging to the family Fabaceae with the very high wood density; ( g ) Averaged minimum monthly precipitation (low=green, high=orange). The north-west everwet Amazon is distinguished by lacking a dry season (precipitation exceeds evapotranspiration). ( h ) Community-weighted wood density 34 . Panels a-d are used as ecotope predictors in the GAM analysis of Supplementary Table 1 . (Data sources: see the ‘Climate variables’ and ‘Climate anomalies for drought definition and mapping’ sections of methods ).

Extended Data Fig. 3 Spatial distributions of climate dynamics in the 2005, 2010 and 2015 droughts.

a – i , Spatial distributions of climate dynamics in the 2005 (left column), 2010 (middle column) and 2015 (right column) droughts for: (a)-(i): Drought dynamics showing drought onset date (row 1, a-c), drought end date (row 2, d-f), and drought duration (row 3, g-i, end date minus start date). Pixel-by-pixel drought responses (EVI in Figs. 1 – 4 ; or GOSIF in Extended Data Figs. 1 & 5 ) are taken as the standardized anomalies that occur during the pixel-specific drought period defined here. (j)-(r): climatic anomalies of: photosynthetic active radiation (PAR) (row 4, j-l), vapor pressure deficit (VPD) (row 5, m-o), and precipitation (row 6, p-r). precipitation (Data source: see the ‘Climate variables’ section of  methods ).

Extended Data Fig. 4 Regions in the Amazon basin.

that emerge from a principal components analysis (PCA) followed by classification: ( a ) PCA of the Amazon basin 0.4° x 0.4° pixel data (coloured according to a supervised classification into three classes identified by variance minimization), projected onto their first two principal components, which are composed mainly of three dimensions, one defined by wood density and proportions of the family Fabaceae (first principal component, horizontal axis), one defined by minimum monthly precipitation and MCWD variability (second principal component, vertical axis), and a third defined mainly by soil fertility; the classes are significantly separated in PCA space (psuedo-F ratio =950, df=2, 3805, p ~ 0, permanova test); ( b ) The Amazon pixels coloured according to their class (corresponding to the colours in a), showing that the classification of (a) maps pixels into distinct, mostly contiguous spatial regions.) ( c ) Standardized values, for each region, of each group of characteristics (ordered by water availability, soil fertility, and tree traits/characteristics), illustrate distinct regional niches: the everwet Amazon is highest in minimum precipitation and lowest (highest negative) in MCWD variability; the Southern Amazon is moderately high in mean fertility, and the Guiana Shield has the tallest mean forest height and greatest wood density. ( d ) scree plot of the eigenvalues (principal components) of the PCA shown in (a), plotted in rank order.

Extended Data Fig. 5 Amazon forest drought responses in different regions using the EVI and GOSIF remote sensing indices.

Amazon forest EVI (top row) and GOSIF (bottom row) responses to multiple droughts in the Guiana shield (left column) and the ever-wet northwest (right column). These generally do not support the “other side of drought” hypothesis 1, because they show generally consistently positive slopes with water-table depth (HAND), in contrast to negative slope responses in the Southern Amazon (Fig. 3a ). Plots show observations (bin average points ±95% CI, and solid regression lines) and unified multi-drought GAM predictions (±95% CI shaded region, for models in Supplementary Table 1b, c ), with climate fixed to region-wide median drought conditions for each drought.) Observations for EVI (a-b): N = 83 and 666 0.4° pixels for 2005 and 2015 droughts respectively, in the Guiana shield (a), and N = 147, 368, and 648 for 2005, 2010 and 2015 droughts respectively in the ever-wet Amazon (b). Observations for GOSIF (c-d): N = 1876, and 25,460 5-km pixels for 2005 and 2015 droughts, respectively, in Guiana shield (c), and N = 1,914, 8,261, and 19,918 for 2005, 2010 and 2015 droughts, respectively, in the ever-wet Amazon (d). Purple points (2010) are not shown in panels a,c, because the 2010 drought did not significantly affect the Guiana shield.

Extended Data Fig. 6 Implementing Structured Causal Modeling (SCM) of Amazon forest drought response using Directed acyclic graphs (DAGs).

a – d , Development of a Directed acyclic graph (DAG) representing the structure of factors influencing tropical forest responses to drought. ( a ) Initially hypothesized DAG characterizing the causal relationships among climatic, environmental, and forest variables (measured variables depicted as blue nodes, unmeasured rooting depth is depicted in grey) leading to forest drought response (other colour node), with arrows representing the hypothesized causal links. ( b ) DAG-data consistency tests for initial DAG , with the largest 20 approximated non-linear correlation coefficients (estimated via root mean square error of approximation, RMSEA) between unlinked variables in (a). (Note: unlinked variables in a DAG are hypothesized to have zero correlation or zero conditional correlation; thus, the second row of panel b tests “DR_ | | _DSL | DL” -- whether DR is independent of DSL conditioned on DL, by estimating the non-linear correlation between DR and the residuals of DSL regressed on DL.) Correlations greater than an acceptability threshold (dashed vertical lines at ±0.30) fail the test of conditional independence, addressed by adding to the DAG either a direct causal link (indicated by a green symbol), or links to a common cause (pink symbol) (such added arrows are included in panel c). ( c ) Final DAG after correcting for conditional independency inconsistencies of the initial DAG in A, in light of ecological considerations. Also illustrates use of the backdoor criterion to determine the causal effect of ‘drought length (DL)’ (the exposed predictor node and associated forward causal paths, in green) on forest drought response (corresponding to the model in Extended Data Fig. 10c ), while blocking the confounding variable dry season length, DSL (hypothesized to itself affect DL) and its associated causal backdoor paths (which are considered non-causal paths with respect to the exposed variable DL) (in pink). ( d ) DAG-Data consistency tests for final DAG (panel c), showing the largest 20 RMSEA values. (e)-(j): GAM regression model predictions (±95% CI shaded region) of causal effects of different variables derived from DAG, employing backdoor criterion, for the Southern Amazon, average across all three droughts: ( e ) of HAND (no backdoor to be blocked) (f) of PAR (adjusting for back door paths through drought length, dry season length) ( g ) of Drought length (adjusting for back door path through dry season length) on EVI responses (adjusted EVI prediction) ; the whole Amazon basin during the 2015 drought: ( h ) of forest height, categorized by shallow (blue, HAND = 0-10 m) and deep (red, HAND = 20–40 m) water tables (adjusting for back door paths through soil fertility, soil texture and dry season length), ( i ) of soil fertility (adjusting for back door path through dry season length) ( j ) of soil texture (no backdoor path to be blocked).

Extended Data Fig. 7 The sensitivities of forest drought response to soil texture and drought timing.

( a ) The sensitivity of forest response to soil texture (sand content) and water- table depth (HAND) in basin-wide GAM analysis. GAM-predicted adjusted EVI anomaly (left axis) versus soil sand content (%), with water table-depth in colour (shallow=blue to deep=red), paired with distributions of mean forest height in each soil texture bin (bottom graph, right axis, with N = 3,318, and 1,142 0.4° pixels for shallow and deep water tables, respectively). ‘Adjusted’ GAM predictions are made by setting non-displayed predictors (climate variables, tree-height, soil fertility) to their median values during the drought. (b)-(d): The sensitivity of forest responses to dry versus wet season drought periods, across the three-droughts: ( b ) distribution of the proportion of drought that was in the dry season (0 = all in the wet season to 1= all in the dry season) for drought-affected pixels in each of the three droughts; ( c ) GAM-predicted EVI anomaly versus PAR, for different proportions of dry season drought (blue=all wet to red=all dry, corresponding to coloured tick marks in the vertical axis of b). ( d ) Adjusted EVI anomaly from GAM prediction versus drought length, for different proportions of dry-season drought (blue to red, as in panel c).

Extended Data Fig. 8 Scale-dependence of Southern Amazon forest responses to drought, showing that detected response patterns are largely invariant across different scales of analysis.

( a ) At 0.4 degree (40-km) scale (across the Southern Amazon. all three droughts): Climate-adjusted EVI responses (standardized anomalies from MODIS) vs. water-table depths (indexed by HAND) for observations (solid points ±95% CI and solid regression line) and for unified multi-drought GAM predictions (model of Supplementary Table 1a , shaded bands and dashed regression line slopes) for the 2005 (green, slope = −0.019 ± 0.001 SE m −1 ), 2010 (purple, slope = −0.020 ± 0.002 SE m −1 ), and 2015 (blue, slope = −0.028 ± 0.002 SE m −1 ) droughts (with N = 1,384, 1,673, and 1,837 0.4° pixels for 2005, 2010, and 2015 droughts, respectively); ( b ) At 1-km scale (across the Southern Amazon, all three droughts), as in (a): climate-adjusted EVI responses vs. HAND for observations (solid points and regression line) and corresponding GAM (with the same Supplementary Table 1a model now fit at 1 km scale, revealing autocorrelation in observations causing too-narrow confidence bands, and slight model underpredictions of the extremes of the 2005 greenup and the 2010 browdown, but maintaining the similar negative dependence on HAND across all droughts); ( c ) At 30 to 180 m scales (for a forest region around Manaus, 2015-2016 drought only): Delta EVI, the fraction change in EVI due to the drought = (after-drought EVI (July 2016) - pre-drought EVI (August 2015))/pre-drought EVI (Landsat OLI8, at 30 m resolution) vs. water-table depths (indexed by HAND) for Landsat observations (solid points ±95% CI and solid regression line) at native (30 m) and aggregated to 90 and 180-m scales (with N = 105,359, 11,901, and 2,999 pixels for 30-m, 90-m, and 180-m scales, respectively). Also shown in the bottom of each panel is the distribution of water-table depth (HAND proxy) at each scale. Aggregations to larger (coarser) scales induce an apparent regression towards the mean in the water-table depth distributions (as more extreme water-table depths at finer scales become diluted by averaging to large scales), while similar dilution of extremes in EVI response (not shown) preserves the overall relation between EVI responses and watertable depth (especially evident in the Landsat analysis where the slopes through data aggregated at different scales do not detectably differ).

Extended Data Fig. 9 Remote sensing validation with forest inventory plot demography.

( a ) Remotely sensed map of MAIAC EVI (1-km resolution), overlaid with aboveground NPP (ANPP) rates from 321 ground-monitored forest plots (red circles, % standing biomass y −1 ) as aggregated to 1 degree grid plots (RAINFOR plots in Brienen et al. 2 ), with both EVI and ANPP taken during the 2000–2011 interval. ANPP rate is calculated as Aboveground Biomass (AGB) gain (Mg/(ha·yr)) (total annual AGB productivity of surviving trees plus recruitment, plus inferred growth of trees that died between censusing intervals) divided by initial AGB (Mg/ha) (standing above ground biomass at the start of the census interval). ( b ) ANPP rates as predicted by EVI (points from (a) plus solid regression line with statistics; Dashed line and associated statistics in grey represent linear regression without the high leverage point, shown in red, defined by Cook’s distances > 4/n, where n=number of points 134 ). EVI is the mean extracted from intervals matching the average census interval of the corresponding plots in Brienen et al. 2 (c)–(e) MAIAC EVI anomalies (1-km pixels) versus ground-monitored tree demography in shallow water table forests during the 2015-2016 drought 26 for : (c) mortality, (d) recruitment, and (e) mortality:recruitment ratios in 1-ha plots. (f)–(h): GOSIF anomalies (5-km pixels) versus ground-monitored (f) mortality, (g) recruitment, and (h) mortality:recruitment ratios; Solid lines and statistics (R 2 and p-values) represent standard linear regression fits to all data. Red points, if they exist, are high leverage, i.e. with Cook’s distances > 4/n, where n=number of points 134 , and dotted lines and associated statistics in grey represent standard linear regressions without such points, showing that remote detection of ground-derived demographic trends is robust. R 2 values reported here are consistent with the expectation that they should be less than for remote detection of tropical forest GPP (R 2  = 0.5-0.7), because GPP contributes only partially to the NPP driver of demography (as discussed in the 'Validation by forest plot metrics of demography and of physiological drought tolerance' section of Methods). Considering multple comparisons (six regressions), the probability, under the null hypothesis, of seeing five or more significant regresssions out of six is p = 0.000002 (Binomial test).

Extended Data Fig. 10 Modeled forest response to the 2015 drought and implications of the derived map of Amazon forest biogeography.

a – c , Forest response to the 2015 drought in drought-affected pixels. ( a ) Observed EVI anomalies (resampled at 0.4 degrees to match model resolution which accounts for spatial autocorrelation (see Supplementary Fig. 1 ). ( b ) GAM-predicted EVI anomalies (model of Supplementary Table 1d ). ( c ) Residual EVI anomalies (panel a observations minus panel b predictions). The GAM well-predicts the pattern of response (Panel b), but under-estimates the extremes of the responses (as evident from residuals in panel c continuing to show greening/browning patterns beyond the predictions). ( d ) Map of Amazon forest biogeography of resilience/vulnerability, overlaid with mean winds (arrows, at height 650 hPa) and location of the arc of deforestation . The most productive as well as the most vulnerable forests (in red) are also those most experiencing deforestation (in the “arc of deforestation”) which is causing local climatic warming/drying 4 , further stressing these vulnerable forests. These “arc of deforestation”/vulnerable forests are often upwind forests 135 (especially when the Intertropical convergence zone, ITCZ, swings to the south) that are critical for hydrological recycling in the Amazon.

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Chen, S., Stark, S.C., Nobre, A.D. et al. Amazon forest biogeography predicts resilience and vulnerability to drought. Nature (2024). https://doi.org/10.1038/s41586-024-07568-w

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