one tailed research hypothesis example

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Statistics and probability

Course: statistics and probability   >   unit 12.

• Hypothesis testing and p-values

One-tailed and two-tailed tests

• Z-statistics vs. T-statistics
• Small sample hypothesis test
• Large sample proportion hypothesis testing

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Video transcript

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What Is a One-Tailed Test?

• Determining Significance
• One-Tailed Test FAQs
• Corporate Finance
• Financial Analysis

One-Tailed Test Explained: Definition and Example

Investopedia / Xiaojie Liu

A one-tailed test is a statistical test in which the critical area of a distribution is one-sided so that it is either greater than or less than a certain value, but not both. If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis.

Financial analysts use the one-tailed test to test an investment or portfolio hypothesis.

Key Takeaways

• A one-tailed test is a statistical hypothesis test set up to show that the sample mean would be higher or lower than the population mean, but not both.
• When using a one-tailed test, the analyst is testing for the possibility of the relationship in one direction of interest and completely disregarding the possibility of a relationship in another direction.
• Before running a one-tailed test, the analyst must set up a null and alternative hypothesis and establish a probability value (p-value).

A basic concept in inferential statistics is hypothesis testing . Hypothesis testing is run to determine whether a claim is true or not, given a population parameter. A test that is conducted to show whether the mean of the sample is significantly greater than and significantly less than the mean of a population is considered a two-tailed test . When the testing is set up to show that the sample mean would be higher or lower than the population mean, it is referred to as a one-tailed test. The one-tailed test gets its name from testing the area under one of the tails (sides) of a normal distribution , although the test can be used in other non-normal distributions.

Before the one-tailed test can be performed, null and alternative hypotheses must be established. A null hypothesis is a claim that the researcher hopes to reject. An alternative hypothesis is the claim supported by rejecting the null hypothesis.

A one-tailed test is also known as a directional hypothesis or directional test.

Example of the One-Tailed Test

Let's say an analyst wants to prove that a portfolio manager outperformed the S&P 500 index in a given year by 16.91%. They may set up the null (H 0 ) and alternative (H a ) hypotheses as:

H 0 : μ ≤ 16.91

H a : μ > 16.91

The null hypothesis is the measurement that the analyst hopes to reject. The alternative hypothesis is the claim made by the analyst that the portfolio manager performed better than the S&P 500. If the outcome of the one-tailed test results in rejecting the null, the alternative hypothesis will be supported. On the other hand, if the outcome of the test fails to reject the null, the analyst may carry out further analysis and investigation into the portfolio manager’s performance.

The region of rejection is on only one side of the sampling distribution in a one-tailed test. To determine how the portfolio’s return on investment compares to the market index, the analyst must run an upper-tailed significance test in which extreme values fall in the upper tail (right side) of the normal distribution curve. The one-tailed test conducted in the upper or right tail area of the curve will show the analyst how much higher the portfolio return is than the index return and whether the difference is significant.

1%, 5% or 10%

The most common significance levels (p-values) used in a one-tailed test.

Determining Significance in a One-Tailed Test

To determine how significant the difference in returns is, a significance level must be specified. The significance level is almost always represented by the letter p, which stands for probability. The level of significance is the probability of incorrectly concluding that the null hypothesis is false. The significance value used in a one-tailed test is either 1%, 5%, or 10%, although any other probability measurement can be used at the discretion of the analyst or statistician. The probability value is calculated with the assumption that the null hypothesis is true. The lower the p-value , the stronger the evidence that the null hypothesis is false.

If the resulting p-value is less than 5%, the difference between both observations is statistically significant, and the null hypothesis is rejected. Following our example above, if the p-value = 0.03, or 3%, then the analyst can be 97% confident that the portfolio returns did not equal or fall below the return of the market for the year. They will, therefore, reject H 0  and support the claim that the portfolio manager outperformed the index. The probability calculated in only one tail of a distribution is half the probability of a two-tailed distribution if similar measurements were tested using both hypothesis testing tools.

When using a one-tailed test, the analyst is testing for the possibility of the relationship in one direction of interest and completely disregarding the possibility of a relationship in another direction. Using our example above, the analyst is interested in whether a portfolio’s return is greater than the market’s. In this case, they do not need to statistically account for a situation in which the portfolio manager underperformed the S&P 500 index. For this reason, a one-tailed test is only appropriate when it is not important to test the outcome at the other end of a distribution.

How Do You Determine If It Is a One-Tailed or Two-Tailed Test?

A one-tailed test looks for an increase or decrease in a parameter. A two-tailed test looks for change, which could be a decrease or an increase.

What Is a One-Tailed T Test Used for?

A one-tailed T-test checks for the possibility of a one-direction relationship but does not consider a directional relationship in another direction.

When Should a Two-Tailed Test Be Used?

You would use a two-tailed test when you want to test your hypothesis in both directions.

University of Southern California. " FAQ: What Are the Differences Between One-Tailed and Two-Tailed Tests? "

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Research Hypothesis In Psychology: Types, & Examples

Saul McLeod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul McLeod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Learn about our Editorial Process

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A research hypothesis, in its plural form “hypotheses,” is a specific, testable prediction about the anticipated results of a study, established at its outset. It is a key component of the scientific method .

Hypotheses connect theory to data and guide the research process towards expanding scientific understanding

Some key points about hypotheses:

• A hypothesis expresses an expected pattern or relationship. It connects the variables under investigation.
• It is stated in clear, precise terms before any data collection or analysis occurs. This makes the hypothesis testable.
• A hypothesis must be falsifiable. It should be possible, even if unlikely in practice, to collect data that disconfirms rather than supports the hypothesis.
• Hypotheses guide research. Scientists design studies to explicitly evaluate hypotheses about how nature works.
• For a hypothesis to be valid, it must be testable against empirical evidence. The evidence can then confirm or disprove the testable predictions.
• Hypotheses are informed by background knowledge and observation, but go beyond what is already known to propose an explanation of how or why something occurs.

Predictions typically arise from a thorough knowledge of the research literature, curiosity about real-world problems or implications, and integrating this to advance theory. They build on existing literature while providing new insight.

Types of Research Hypotheses

Alternative hypothesis.

The research hypothesis is often called the alternative or experimental hypothesis in experimental research.

It typically suggests a potential relationship between two key variables: the independent variable, which the researcher manipulates, and the dependent variable, which is measured based on those changes.

The alternative hypothesis states a relationship exists between the two variables being studied (one variable affects the other).

A hypothesis is a testable statement or prediction about the relationship between two or more variables. It is a key component of the scientific method. Some key points about hypotheses:

• Important hypotheses lead to predictions that can be tested empirically. The evidence can then confirm or disprove the testable predictions.

In summary, a hypothesis is a precise, testable statement of what researchers expect to happen in a study and why. Hypotheses connect theory to data and guide the research process towards expanding scientific understanding.

An experimental hypothesis predicts what change(s) will occur in the dependent variable when the independent variable is manipulated.

It states that the results are not due to chance and are significant in supporting the theory being investigated.

The alternative hypothesis can be directional, indicating a specific direction of the effect, or non-directional, suggesting a difference without specifying its nature. It’s what researchers aim to support or demonstrate through their study.

Null Hypothesis

The null hypothesis states no relationship exists between the two variables being studied (one variable does not affect the other). There will be no changes in the dependent variable due to manipulating the independent variable.

It states results are due to chance and are not significant in supporting the idea being investigated.

The null hypothesis, positing no effect or relationship, is a foundational contrast to the research hypothesis in scientific inquiry. It establishes a baseline for statistical testing, promoting objectivity by initiating research from a neutral stance.

Many statistical methods are tailored to test the null hypothesis, determining the likelihood of observed results if no true effect exists.

This dual-hypothesis approach provides clarity, ensuring that research intentions are explicit, and fosters consistency across scientific studies, enhancing the standardization and interpretability of research outcomes.

Nondirectional Hypothesis

A non-directional hypothesis, also known as a two-tailed hypothesis, predicts that there is a difference or relationship between two variables but does not specify the direction of this relationship.

It merely indicates that a change or effect will occur without predicting which group will have higher or lower values.

For example, “There is a difference in performance between Group A and Group B” is a non-directional hypothesis.

Directional Hypothesis

A directional (one-tailed) hypothesis predicts the nature of the effect of the independent variable on the dependent variable. It predicts in which direction the change will take place. (i.e., greater, smaller, less, more)

It specifies whether one variable is greater, lesser, or different from another, rather than just indicating that there’s a difference without specifying its nature.

For example, “Exercise increases weight loss” is a directional hypothesis.

Falsifiability

The Falsification Principle, proposed by Karl Popper , is a way of demarcating science from non-science. It suggests that for a theory or hypothesis to be considered scientific, it must be testable and irrefutable.

Falsifiability emphasizes that scientific claims shouldn’t just be confirmable but should also have the potential to be proven wrong.

It means that there should exist some potential evidence or experiment that could prove the proposition false.

However many confirming instances exist for a theory, it only takes one counter observation to falsify it. For example, the hypothesis that “all swans are white,” can be falsified by observing a black swan.

For Popper, science should attempt to disprove a theory rather than attempt to continually provide evidence to support a research hypothesis.

Can a Hypothesis be Proven?

Hypotheses make probabilistic predictions. They state the expected outcome if a particular relationship exists. However, a study result supporting a hypothesis does not definitively prove it is true.

All studies have limitations. There may be unknown confounding factors or issues that limit the certainty of conclusions. Additional studies may yield different results.

In science, hypotheses can realistically only be supported with some degree of confidence, not proven. The process of science is to incrementally accumulate evidence for and against hypothesized relationships in an ongoing pursuit of better models and explanations that best fit the empirical data. But hypotheses remain open to revision and rejection if that is where the evidence leads.

• Disproving a hypothesis is definitive. Solid disconfirmatory evidence will falsify a hypothesis and require altering or discarding it based on the evidence.
• However, confirming evidence is always open to revision. Other explanations may account for the same results, and additional or contradictory evidence may emerge over time.

We can never 100% prove the alternative hypothesis. Instead, we see if we can disprove, or reject the null hypothesis.

If we reject the null hypothesis, this doesn’t mean that our alternative hypothesis is correct but does support the alternative/experimental hypothesis.

Upon analysis of the results, an alternative hypothesis can be rejected or supported, but it can never be proven to be correct. We must avoid any reference to results proving a theory as this implies 100% certainty, and there is always a chance that evidence may exist which could refute a theory.

How to Write a Hypothesis

• Identify variables . The researcher manipulates the independent variable and the dependent variable is the measured outcome.
• Operationalized the variables being investigated . Operationalization of a hypothesis refers to the process of making the variables physically measurable or testable, e.g. if you are about to study aggression, you might count the number of punches given by participants.
• Decide on a direction for your prediction . If there is evidence in the literature to support a specific effect of the independent variable on the dependent variable, write a directional (one-tailed) hypothesis. If there are limited or ambiguous findings in the literature regarding the effect of the independent variable on the dependent variable, write a non-directional (two-tailed) hypothesis.
• Make it Testable : Ensure your hypothesis can be tested through experimentation or observation. It should be possible to prove it false (principle of falsifiability).
• Clear & concise language . A strong hypothesis is concise (typically one to two sentences long), and formulated using clear and straightforward language, ensuring it’s easily understood and testable.

Consider a hypothesis many teachers might subscribe to: students work better on Monday morning than on Friday afternoon (IV=Day, DV= Standard of work).

Now, if we decide to study this by giving the same group of students a lesson on a Monday morning and a Friday afternoon and then measuring their immediate recall of the material covered in each session, we would end up with the following:

• The alternative hypothesis states that students will recall significantly more information on a Monday morning than on a Friday afternoon.
• The null hypothesis states that there will be no significant difference in the amount recalled on a Monday morning compared to a Friday afternoon. Any difference will be due to chance or confounding factors.

More Examples

• Memory : Participants exposed to classical music during study sessions will recall more items from a list than those who studied in silence.
• Social Psychology : Individuals who frequently engage in social media use will report higher levels of perceived social isolation compared to those who use it infrequently.
• Developmental Psychology : Children who engage in regular imaginative play have better problem-solving skills than those who don’t.
• Clinical Psychology : Cognitive-behavioral therapy will be more effective in reducing symptoms of anxiety over a 6-month period compared to traditional talk therapy.
• Cognitive Psychology : Individuals who multitask between various electronic devices will have shorter attention spans on focused tasks than those who single-task.
• Health Psychology : Patients who practice mindfulness meditation will experience lower levels of chronic pain compared to those who don’t meditate.
• Organizational Psychology : Employees in open-plan offices will report higher levels of stress than those in private offices.
• Behavioral Psychology : Rats rewarded with food after pressing a lever will press it more frequently than rats who receive no reward.

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Statistics By Jim

Making statistics intuitive

When Can I Use One-Tailed Hypothesis Tests?

By Jim Frost 16 Comments

One-tailed hypothesis tests offer the promise of more statistical power compared to an equivalent two-tailed design. While there is some debate about when you can use a one-tailed test, the general consensus among statisticians is that you should use two-tailed tests unless you have concrete reasons for using a one-tailed test.

In this post, I discuss when you should and should not use one-tailed tests. I’ll cover the different schools of thought and offer my own opinion.

If you need to learn the basics about these two types of test, please read my previous post: One-Tailed and Two-Tailed Hypothesis Tests Explained .

Two-Tailed Tests are the Default Choice

The vast majority of hypothesis tests that analysts perform are two-tailed because they can detect effects in both directions. This fact is generally the clincher. In most studies, you are interested in determining whether there is a positive effect or a negative effect. In other words, results in either direction provide essential information. If this statement describes your study, you must use a two-tailed test. There’s no need to read any further. Typically, you need a strong reason to move away from using two-tailed tests.

On the other hand, there are some cases where one-tailed tests are not only a valid option, but truly are a requirement.

Consequently, there is a spectrum that ranges from cases where one-tailed tests are definitely not appropriate to cases where they are required. In the middle of this spectrum, there are cases where analysts might disagree. The breadth of opinions extends from those who believe you should use one-tailed tests for only a few specific situations when they are required to those who are more lenient about their usage.

A Concrete Rule about Choosing Between One- and Two-Tailed Tests

Despite this disagreement, there is a hard and fast rule about the decision process itself upon which all statisticians agree. You must decide whether you will use a one-tailed or two-tailed test at the beginning of your study before you look at your data. You must not perform a two-tailed analysis, obtain non-significant results, and then try a one-tailed test to see if that is statistically significant. If you plan to use a one-tailed test, make this decision at the beginning of the study and explain why it is the proper choice.

The approach I take is to assume you’ll use a two-tailed test and then move away from that only after carefully determining that a one-tailed test is appropriate for your study. The following are potential reasons for why you might use a one-tailed hypothesis test.

Related post : 5 Steps for Conducting Scientific Studies with Statistical Analyses

One-Tailed Tests Can Be the Only Option

For some hypothesis tests, the mechanics of how a test functions dictate using a one-tailed methodology. Chi-squared tests and F-tests and are often one-tailed for this reason.

Chi-squared tests

Analysts often use chi-squared tests to determine whether data fit a theoretical distribution and whether categorical variables are independent . For these tests, when the chi-squared value exceeds the critical threshold, you have sufficient evidence to conclude that the data do not follow the distribution or that the categorical variables are dependent. The chi-squared value either reaches this threshold or it does not. For all values below the threshold, you fail to reject the null hypothesis. There is no other interpretation for very low chi-squared values. Hence, these tests are one-tailed by their nature.

F-tests are highly flexible tests that analysts use in a wide variety of scenarios. Some of these scenarios exclude the possibility of a two-tailed test. For instance, F-tests in ANOVA and the overall test of significance for linear models are similar to the chi-squared example. The F ratio can increase to the significance threshold or it does not. In one-way ANOVA, if the F-value surpasses the threshold, you can conclude that not all group means are equal. On the other hand, all F-values below the threshold yield the same interpretation—the sample provides insufficient evidence to conclude that the group means are unequal. No other effect or interpretation exists for very low F-values.

When a one-tailed version of the test is the only meaningful possibility, statistical software won’t ask you to make a choose. That’s why you’ll never need to choose between a one or two-tailed ANOVA F-test or chi-square tests.

In some cases, the nature of the test itself requires using a one-sided methodology, and it does not depend on the study area.

Effects can Occur in Only One Direction

On the other hand, other hypothesis tests can legitimately have one and two-tailed versions, and you need to choose between them based on the study area. Tests that fall in this category include t-tests , proportion tests, Poisson rate tests, variance tests, and some nonparametric tests for the median. In these cases, base the decision on subject-area knowledge about the possible effects.

For some study areas, the effect can exist in only one direction. It simply can’t exist in the other direction. To make this determination, you need to use your subject-area knowledge and understanding of physical limitations. In this case, if there were a difference in the untested direction, you would attribute it to random error regardless of how large it is. In other words, only chance can produce an observed effect in the other direction. If you have even the smallest notion that an observed effect in the other direction could be a real effect rather than random error, use a two-tailed test.

For example, imagine we are comparing an herbicide’s ability to kill weeds to no treatment. We randomly apply the herbicide to some patches of grass and no herbicide to other patches. It is inconceivable that the herbicide can promote weed growth. In the worst-case scenario, it is entirely ineffective, and the herbicide patches should be equivalent to the control group. If the herbicide patches ultimately have more weeds than the control group, we’ll chalk that up to random error regardless of the difference—even if it’s substantial. In this case, we are wholly justified using a one-tailed test to determine whether the herbicide is better than no treatment.

No Controversy So Far!

So far, the preceding two reasons fall entirely on safe ground. Using one-tailed tests because of its mechanics or because an effect can occur in only one direction should be acceptable to all statisticians. In fact, some statisticians believe that these are the only valid reasons for using one-tailed hypothesis tests. I happen to fall within this school of thought myself.

In the next section, I’ll discuss a scenario where some analysts believe you can choose between one and two-tailed tests, but others disagree with that notion.

You Only Need to Know About Effects in One Direction

In this scenario, effects can exist in both directions, but you only care about detecting an effect in one direction. Analysts use the one-tailed approach in this situation to boost the statistical power of the hypothesis test .

To even consider using a one-tailed test for this reason, you must be entirely sure there is no need to detect an effect in the other direction. While you gain more statistical power in one direction, the test has absolutely no power in the other direction.

Suppose you are testing a new vaccine and want to determine whether it’s better than the current vaccine. You use a one-tailed test to improve the test’s ability to learn whether the new vaccine is better. However, that’s unethical because the test cannot determine whether it is less effective. You risk missing valuable information by testing in only one direction.

However, there might be occasions where you, or science, genuinely don’t need to detect an effect in the untested direction. For example, suppose you are considering a new part that is cheaper than the current part. Your primary motivation for switching is the price reduction. The new part doesn’t have to be better than the current part, but it cannot be worse. In this case, it might be appropriate to perform a one-tailed test that determines whether the new part is worse than the old part. You won’t know if it is better, but you don’t need to know that.

As I mentioned, many statisticians don’t think you should use a one-tailed test for this type of scenario. My position is that you should set up a two-tailed test that produces the same power benefits as a one-tailed test because that approach will accurately capture the underlying fact that effects can occur in both directions.

However, before explaining this alternate approach, I need to describe an additional problem with the above scenario.

Beware of the Power that One-Tailed Tests Provide

The promise of extra statistical power in the direction of interest is tempting. After all, if you don’t care about effects in the opposite direction, what’s the problem? It turns out there is an additional penalty that comes with the extra power.

First, let’s see why one-tailed tests are more powerful than two-tailed tests with the same significance level . The graphs below display the t-distributions for two t-tests with the same sample size. I show the critical t-values for both tests. As you can see, the one-tailed test requires a less extreme t-value (1.725) to produce a statistically significant result in the right tail than the two-tailed test (2.086). In other words, a smaller effect is statistically significant in the one-tailed test.

Both tests have the same Type I error rate because we defined the significance level as 0.05. This type of error occurs when the test rejects a true null hypothesis—a false positive. This error rate corresponds to the total percentage of the shaded areas under the curve. While both tests have the same overall Type I error rate, the distribution of these errors is different.

To understand why, keep in mind that the critical regions also represent where the Type I errors occur. For a two-tailed test, these errors are split equally between the left and right tails. However, for a one-tailed test, all of these errors arise specifically in the one direction that you are interested in. Unfortunately, the error rate doubles in that direction compared to a two-tailed test. In the graphs above, the right tail has an error rate of 5% in the one-tailed test compared to 2.5% in the two-tailed test.

Related Post : Types of Errors in Hypothesis Tests

You Haven’t Changed Anything of Substance

By switching to a one-tailed test, you haven’t changed anything of substance to gain this extra power. All you’ve done is to redraw the critical region so that a smaller effect in the direction of interest is statistically significant. In this light, it’s not surprising that merely labeling smaller effects as being statistically significant also produces more false positives in that direction! And, the graphs reflect that fact.

If you want to increase the test’s power without increasing the Type I error rate, you’ll need to make a more fundamental change to your study’s design, such as increasing your sample size or more effectively controlling the variability.

Is the Higher False Positive Rate Worthwhile?

To use a one-tailed test to gain more power, you can’t care about detecting an effect in the other direction, and you have to be willing to accept twice the false positives in the direction you are interested. Remember, a false positive means that you will not obtain the benefits you expect.

Should you accept double the false positives in the direction of interest? Answering that question depends on the actions that a significant result will prompt. If you’re considering changing to a new production line, that’s a very costly decision. Doubling the false positives is problematic. Your company will spend a lot of money for a new manufacturing line, but it might not produce better products. However, if you’re changing suppliers for a part based on the test result, and their parts don’t cost more, a false positive isn’t an expensive problem.

Think carefully about whether the additional power is worth the extra false positives in your direction of interest! If you decide that the added power is worth the risk, consider my alternative approach below. It produces an equivalent amount of statistical power as the one-tailed approach. However, it uses a methodology that more accurately reflects the underlying reality of the study area and the goals of the analyst.

Alternative: Use a Two-Tailed Test with a Higher Significance Level

In my view, determining the possible directions of an effect and the statistical power of the analysis are two independent issues. Using a one-tailed test to boost power can obscure these matters and their ramifications. My recommendation is to use the following process:

• Identify the directions that an effect can occur, and then choose a one-tailed or two-tailed test accordingly.
• Choose the significance level to correctly set the sensitivity and false-positive rate based on your specific requirements.

This process breaks down the questions you need to answer into two separate issues, which allows you to consider each more carefully.

Now, let’s apply this process to the scenario where you’re studying an effect that can occur in both directions, but the following are both true:

• You care about effects in only one direction.
• Increasing the power of the test is worth a higher risk of false positives in that direction.

In this situation, using a one-tailed test to gain extra power seems like an acceptable solution. However, that approach attempts to solve the right problem by using the wrong methodology. Here’s my alternative method.

Instead of using a one-tailed test, consider using a two-tailed test and doubling the significance level, such as from 0.05 to 0.10. This approach increases your power while allowing the test methodology to match the reality of the situation better. It also increases the transparency of your goals as the analyst.

Related Post : Significance Levels and P-values

How the Two-Tailed Approach with a Higher Significance Level Works

To understand this approach, compare the graphs below. The top graph is one-sided and uses a significance level of 0.05. The bottom graph is two-sided and uses a significance level of 0.10.

As you can see in the graphs, the critical region on the right side of both distributions starts at the same critical t-value (1.725). Consequently, both the one- and two-tailed tests provide the same power in that direction. Additionally, there is a critical region in the other tail, which means that the test can detect effects in the opposite direction as well.

The end result is that the two-tailed test has the same power and an equal probability of a Type I error in the direction of interest. Great! And, you can detect effects in the other direction even though you might not need to know about them. Okay, that’s not a bad thing.

This Approach Is More Transparent

What’s so great about this approach? It makes your methodology choices more explicit while accurately reflecting a study area where effects can occur in both directions. Here’s how.

The significance level is an evidentiary standard for the amount of sample evidence required to reject the null hypothesis. By increasing the significance level from 0.05 to 0.10, you’re explicitly stating that you are lowering the amount of evidence necessary to reject the null, which logically increases the power of the test. Additionally, as you raise the significance level, the Type I error rate also increases by definition. This approach produces the same power gains as a one-tailed test. However, it more clearly indicates how the analyst set up a more sensitive test in exchange for a higher risk of false positives.

The problem with gaining the additional power by switching to a one-tailed test is that it obscures the fact that you’re weakening the evidentiary standard. After all, you’re not explicitly changing the significance level. That’s why the increase in the Type I error rate in the direction of interest can be surprising!

Decision Guidelines

We covered a lot in this post. Here’s a brief recap of when to use each type of test. For some tests, you don’t have to worry about this choice. However, if you do need to decide between using a one-tailed and a two-tailed test, follow these guidelines. If the effect can occur in:

• One direction: Use a one-tailed test and choose the correct alternative hypothesis .
• Both directions: Use a two-tailed test.
• Both directions, but you care about only one direction and you need the higher statistical power: Use a two-tailed test and double the significance level. Be aware that you are doubling the probability of a false positive.

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Reader Interactions

April 13, 2021 at 10:02 am

Thanks Jim!

April 12, 2021 at 1:57 pm

Another great post.

If my hypothesis was say, that intelligence overall will be greater for first group that took the study in 2010 than the second group that took the same test in 2020. Would this be one tailed because I have made a specific prediction about the direction of intelligence over time?

Thanks again, Grace

April 13, 2021 at 12:22 am

I think you’d have a stronger case for a one-tailed test if the studies were closer together in time. When they’re so far apart, it’s possible that intelligence could decline over the years. (I’ve seen it happen!) But, if the studies were say a month apart, you’d have a stronger case for saying that intelligence wouldn’t decline over such a short span of time and, therefore, a one-tailed test might be called for. Whenever you can say that an effect is only possible in one direction, that’s the strongest case for a one-tailed test where you won’t get any debate.

It sounds like you’re asking about a one-tailed test based on a prediction about the hypothesis. That’s not usually a good enough reason to use a one-tailed test by itself. Of course, as I mention, there is some debate about when it’s ok. At the very least, it could be based on your prediction and the fact that you don’t care about results in the other direction. If you wanted to get published in a journal, that wouldn’t fly. Outside the academic context, you’d probably get some analysts to agree with that case and others wouldn’t.

Just be aware of the drawbacks that I mention. By going to a one-tailed tests, you’re doubling the false positives in the hypothesis direction in which you’re interested. I only recommend one-tailed tests for cases where the effect can only possibly exist in one direction.

November 24, 2020 at 12:33 pm

Brilliant post, Jim! I use hypothesis tests all the time (always two-tailed), but with the explanation you provided here, I can raise the significance if more false positives (i.e., Type I errors) are not a problem. With that said, this approach would still have to get past reviewers in a manuscript submission, which is no sure thing. I’ll play with the numbers if this comes up again in my work — and I will read this post at least once more, too. Thanks for the insight.

November 24, 2020 at 10:51 pm

Great to hear from you again! I’m glad this post was helpful. I think typically you wouldn’t want to raise the significance level higher that 0.05. However, for those who change to a one-sided test and leave the significance level at 0.05, they’re doing that in effect.

Best wishes and Happy Thanksgiving!

November 18, 2020 at 6:19 am

November 8, 2020 at 12:49 am

By “not all the data falls within a particular region”, do you mean that some of the data collected fall in the region and others don’t , BUT the mean of all data in this particular sample do, which is the whole point of hypothesis testing? As to the curve, I think that is the hypothesized sampling distribution of the sample mean, with the sample collected being a member of the overall set. please advise whether this is right, if not , then, there really is something wrong with the understanding and I will go back to the text book : ), otherwise, my question regarding one tailed test remains, thank you so much Jim.

November 8, 2020 at 3:35 pm

I do highly recommend that you read the post I link for you. It’ll help!

There seems a crucial piece that you’re missing. Again, it’s totally understandable because it’s not obvious.

These tests don’t assess where individual data points fall in a distribution.

Instead, these tests assess one estimate of a population parameter and compare it to the null hypothesis value.

Let’s look at that in the context of a 1-sample t-test. In this case, you’re comparing the sample mean (which estimates the population mean) and comparing it to the null hypothesis value. So, it’s just one value (the sample mean), not all the data points. And, you’re looking to see where that value falls in relation to the null hypothesis value. In the graphs, the null hypothesis value is the peak. And, you’re looking to see how far out the mean is. And because the mean is only one value, it’ll fall only at one point on the graphs.

Again, read the other post. It’ll answer your questions. It doesn’t make sense for me retype what I wrote in that post here in the comments. If after reading that post you have more questions, I’ll be happy to answer! 🙂

November 7, 2020 at 12:42 am

Use your illustration “Null: The effect is less than or equal to zero. Alternative: The effect is greater than zero.” Say if the significance level is 0.05, all on the left side, we are saying that 5% of the data are in the region, and if we observe this unlikely event, then it’s unlikely that the hypothesized mean is the true mean, depending how rare you want your criteria to be. If say the alpha level is 1%, where the critical value is even further from the mean, and if the p value is still in there, then we can be even more confident.I hope I am right about the above, but even if I am, I am still not as comfortable with “less than” as with “ not equal to”, even though I can work through the mechanics and get most my practice questions right. Is it ok to say that, at most 1% of data is in there, given the distribution, because any means greater will have a lower percentage, so 0.9%, 0.8%, 0.7% etc as you shift your means to the right with no boundaries, so you can shift infinitely, therefore we can be 99% confident that it is less than, please? Or if not, what’s the logic in words please? Thank you.

November 8, 2020 at 12:15 am

That’s not what the significance level indicates. The significance level doesn’t indicate where the data fall. If you’re performing a one-tailed test and get significant results, it doesn’t mean all the data falls in a particular region of the curve. It means that the sample statistic, such as the mean effect, falls far enough away from zero in a particular direction such that the test statistic falls in the corresponding critical region. The curves you’re seeing in the graphs are not data distributions. They’re sampling distributions for the test statistic, which is an entirely different thing.

I think before trying to understand one-tailed tests, you should read more about how hypothesis tests work in general. Click that link to learn more about how they work, sampling distributions, and what significance levels and p-values actually mean. I can tell you have a few misconceptions about them. That’s ok because they’re tricky concepts. But, it’ll be difficult to understand one-tailed tests without fully understanding how hypothesis tests work.

November 5, 2020 at 9:19 am

I can tell that in a two tailed test, the rejection regions are such that only a certain percentage of data points falls within that range and if you happen to observe a data point within that range, then it’s ok to conclude that the hypothesized mean is unlikely the true mean. However, if I shift all the rejection region to one side, knowing how unlikely I will find something in there, and then somehow observe a data within the range, how does it lead to a conclusion that the true mean is greater or smaller than the hypothesized value please? How can I draw any conclusion from this observation ? If the true mean is to either the left or right of the hypothesized value, it will have its own distribution , rendering the existing distribution irrelevant for drawing conclusion about a different mean?

November 6, 2020 at 9:23 pm

To be technically correct, you’re not looking for data points to fall in the critical regions. Instead, you’re looking for sample statistics that fall in those regions. You don’t need to worry about the distribution changing based on whether the mean is greater than or less than. It all works on the same distribution, which is a sampling distribution for the test statistic.

Read my post about one-tailed and two-tailed tests . It’ll show you how they work and I believe will answer your questions. I show the distributions for both types. If you do have more questions after reading that, don’t hesitate to ask!

May 10, 2020 at 5:32 am

Nice article, thank you Mr. Frost. I am a statistician and I run in this problem regularly and I am still not clear with it. With “Both directions but you care about only one direction” I use the approach that I do 2-tail test on 5 % sig. level and if this is significant and my client is interested only in one direction, then I interpret that the one-sided effect is significant at 5 % level. Which may look weird, but it is a correct statement. Basically, I avoid stating that one sided effect is significant at 5 % level in the situation where the 2-sided p-value is e.g. 0.07 and 1-sided is 0.035. This I don’t interpret as significant on 5 % even if my client is interested only in one direction.

December 26, 2018 at 1:20 am

Ye ! This Is A Good Blog!

November 12, 2018 at 10:00 am

Its a good article Mr. Jim. It gives clarication to the one tailed and two tailed tests that we commonly use in research.

November 12, 2018 at 11:00 am

Thanks, Sreekumar! I’m glad it was helpful!

Comments and Questions Cancel reply

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Statistical Methods and Data Analytics

FAQ: What are the differences between one-tailed and two-tailed tests?

When you conduct a test of statistical significance, whether it is from a correlation, an ANOVA, a regression or some other kind of test, you are given a p-value somewhere in the output.  If your test statistic is symmetrically distributed, you can select one of three alternative hypotheses. Two of these correspond to one-tailed tests and one corresponds to a two-tailed test.  However, the p-value presented is (almost always) for a two-tailed test.  But how do you choose which test?  Is the p-value appropriate for your test? And, if it is not, how can you calculate the correct p-value for your test given the p-value in your output?  

What is a two-tailed test?

First let’s start with the meaning of a two-tailed test.  If you are using a significance level of 0.05, a two-tailed test allots half of your alpha to testing the statistical significance in one direction and half of your alpha to testing statistical significance in the other direction.  This means that .025 is in each tail of the distribution of your test statistic. When using a two-tailed test, regardless of the direction of the relationship you hypothesize, you are testing for the possibility of the relationship in both directions.  For example, we may wish to compare the mean of a sample to a given value x using a t-test.  Our null hypothesis is that the mean is equal to x . A two-tailed test will test both if the mean is significantly greater than x and if the mean significantly less than x . The mean is considered significantly different from x if the test statistic is in the top 2.5% or bottom 2.5% of its probability distribution, resulting in a p-value less than 0.05.     

What is a one-tailed test?

Next, let’s discuss the meaning of a one-tailed test.  If you are using a significance level of .05, a one-tailed test allots all of your alpha to testing the statistical significance in the one direction of interest.  This means that .05 is in one tail of the distribution of your test statistic. When using a one-tailed test, you are testing for the possibility of the relationship in one direction and completely disregarding the possibility of a relationship in the other direction.  Let’s return to our example comparing the mean of a sample to a given value x using a t-test.  Our null hypothesis is that the mean is equal to x . A one-tailed test will test either if the mean is significantly greater than x or if the mean is significantly less than x , but not both. Then, depending on the chosen tail, the mean is significantly greater than or less than x if the test statistic is in the top 5% of its probability distribution or bottom 5% of its probability distribution, resulting in a p-value less than 0.05.  The one-tailed test provides more power to detect an effect in one direction by not testing the effect in the other direction. A discussion of when this is an appropriate option follows.   

When is a one-tailed test appropriate?

Because the one-tailed test provides more power to detect an effect, you may be tempted to use a one-tailed test whenever you have a hypothesis about the direction of an effect. Before doing so, consider the consequences of missing an effect in the other direction.  Imagine you have developed a new drug that you believe is an improvement over an existing drug.  You wish to maximize your ability to detect the improvement, so you opt for a one-tailed test. In doing so, you fail to test for the possibility that the new drug is less effective than the existing drug.  The consequences in this example are extreme, but they illustrate a danger of inappropriate use of a one-tailed test.

So when is a one-tailed test appropriate? If you consider the consequences of missing an effect in the untested direction and conclude that they are negligible and in no way irresponsible or unethical, then you can proceed with a one-tailed test. For example, imagine again that you have developed a new drug. It is cheaper than the existing drug and, you believe, no less effective.  In testing this drug, you are only interested in testing if it less effective than the existing drug.  You do not care if it is significantly more effective.  You only wish to show that it is not less effective. In this scenario, a one-tailed test would be appropriate. 

When is a one-tailed test NOT appropriate?

Choosing a one-tailed test for the sole purpose of attaining significance is not appropriate.  Choosing a one-tailed test after running a two-tailed test that failed to reject the null hypothesis is not appropriate, no matter how "close" to significant the two-tailed test was.  Using statistical tests inappropriately can lead to invalid results that are not replicable and highly questionable–a steep price to pay for a significance star in your results table!   

Deriving a one-tailed test from two-tailed output

The default among statistical packages performing tests is to report two-tailed p-values.  Because the most commonly used test statistic distributions (standard normal, Student’s t) are symmetric about zero, most one-tailed p-values can be derived from the two-tailed p-values.   

Below, we have the output from a two-sample t-test in Stata.  The test is comparing the mean male score to the mean female score.  The null hypothesis is that the difference in means is zero.  The two-sided alternative is that the difference in means is not zero.  There are two one-sided alternatives that one could opt to test instead: that the male score is higher than the female score (diff  > 0) or that the female score is higher than the male score (diff < 0).  In this instance, Stata presents results for all three alternatives.  Under the headings Ha: diff < 0 and Ha: diff > 0 are the results for the one-tailed tests. In the middle, under the heading Ha: diff != 0 (which means that the difference is not equal to 0), are the results for the two-tailed test. 

Two-sample t test with equal variances ------------------------------------------------------------------------------ Group | Obs Mean Std. Err. Std. Dev. [95% Conf. Interval] ---------+-------------------------------------------------------------------- male | 91 50.12088 1.080274 10.30516 47.97473 52.26703 female | 109 54.99083 .7790686 8.133715 53.44658 56.53507 ---------+-------------------------------------------------------------------- combined | 200 52.775 .6702372 9.478586 51.45332 54.09668 ---------+-------------------------------------------------------------------- diff | -4.869947 1.304191 -7.441835 -2.298059 ------------------------------------------------------------------------------ Degrees of freedom: 198 Ho: mean(male) - mean(female) = diff = 0 Ha: diff < 0 Ha: diff != 0 Ha: diff > 0 t = -3.7341 t = -3.7341 t = -3.7341 P < t = 0.0001 P > |t| = 0.0002 P > t = 0.9999

Note that the test statistic, -3.7341, is the same for all of these tests.  The two-tailed p-value is P > |t|. This can be rewritten as P(>3.7341) + P(< -3.7341).  Because the t-distribution is symmetric about zero, these two probabilities are equal: P > |t| = 2 *  P(< -3.7341).  Thus, we can see that the two-tailed p-value is twice the one-tailed p-value for the alternative hypothesis that (diff < 0).  The other one-tailed alternative hypothesis has a p-value of P(>-3.7341) = 1-(P<-3.7341) = 1-0.0001 = 0.9999.   So, depending on the direction of the one-tailed hypothesis, its p-value is either 0.5*(two-tailed p-value) or 1-0.5*(two-tailed p-value) if the test statistic symmetrically distributed about zero. 

In this example, the two-tailed p-value suggests rejecting the null hypothesis of no difference. Had we opted for the one-tailed test of (diff > 0), we would fail to reject the null because of our choice of tails. 

The output below is from a regression analysis in Stata.  Unlike the example above, only the two-sided p-values are presented in this output.

Source | SS df MS Number of obs = 200 -------------+------------------------------ F( 2, 197) = 46.58 Model | 7363.62077 2 3681.81039 Prob > F = 0.0000 Residual | 15572.5742 197 79.0486001 R-squared = 0.3210 -------------+------------------------------ Adj R-squared = 0.3142 Total | 22936.195 199 115.257261 Root MSE = 8.8909 ------------------------------------------------------------------------------ socst | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- science | .2191144 .0820323 2.67 0.008 .0573403 .3808885 math | .4778911 .0866945 5.51 0.000 .3069228 .6488594 _cons | 15.88534 3.850786 4.13 0.000 8.291287 23.47939 ------------------------------------------------------------------------------

For each regression coefficient, the tested null hypothesis is that the coefficient is equal to zero.  Thus, the one-tailed alternatives are that the coefficient is greater than zero and that the coefficient is less than zero. To get the p-value for the one-tailed test of the variable science having a coefficient greater than zero, you would divide the .008 by 2, yielding .004 because the effect is going in the predicted direction. This is P(>2.67). If you had made your prediction in the other direction (the opposite direction of the model effect), the p-value would have been 1 – .004 = .996.  This is P(<2.67). For all three p-values, the test statistic is 2.67. 

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One-Tailed Hypothesis Tests: 3 Example Problems

In statistics, we use hypothesis tests to determine whether some claim about a population parameter is true or not.

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

There are two types of hypothesis tests:

• Two-tailed test : Alternative hypothesis contains the ≠ sign
• One-tailed test : Alternative hypothesis contains either or > sign

In a one-tailed test , the alternative hypothesis contains the less than (“”) sign. This indicates that we’re testing whether or not there is a positive or negative effect.

Check out the following example problems to gain a better understanding of one-tailed tests.

Example 1: Factory Widgets

Suppose it’s assumed that the average weight of a certain widget produced at a factory is 20 grams. However, one engineer believes that a new method produces widgets that weigh less than 20 grams.

To test this, he can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

• H 0 (Null Hypothesis): μ ≥ 20 grams
• H A (Alternative Hypothesis): μ

Note : We can tell this is a one-tailed test because the alternative hypothesis contains the less than ( ) sign. Specifically, we would call this a left-tailed test because we’re testing if some population parameter is less than a specific value.

To test this, he uses the new method to produce 20 widgets and obtains the following information:

• n = 20 widgets
• x = 19.8 grams
• s = 3.1 grams

Plugging these values into the One Sample t-test Calculator , we obtain the following results:

• t-test statistic: -0.288525
• one-tailed p-value: 0.388

Since the p-value is not less than .05, the engineer fails to reject the null hypothesis.

He does not have sufficient evidence to say that the true mean weight of widgets produced by the new method is less than 20 grams.

Example 2: Plant Growth

Suppose a standard fertilizer has been shown to cause a species of plants to grow by an average of 10 inches. However, one botanist believes a new fertilizer can cause this species of plants to grow by an average of greater than 10 inches.

To test this, she can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

• H 0 (Null Hypothesis): μ ≤ 10 inches
• H A (Alternative Hypothesis): μ > 10 inches

Note : We can tell this is a one-tailed test because the alternative hypothesis contains the greater than ( > ) sign. Specifically, we would call this a right-tailed test because we’re testing if some population parameter is greater than a specific value.

To test this claim, she applies the new fertilizer to a simple random sample of 15 plants and obtains the following information:

• n = 15 plants
• x = 11.4 inches
• s = 2.5 inches
• t-test statistic: 2.1689
• one-tailed p-value: 0.0239

Since the p-value is less than .05, the botanist rejects the null hypothesis.

She has sufficient evidence to conclude that the new fertilizer causes an average increase of greater than 10 inches.

Example 3: Studying Method

A professor currently teaches students to use a studying method that results in an average exam score of 82. However, he believes a new studying method can produce exam scores with an average value greater than 82.

To test this, he can perform a one-tailed hypothesis test with the following null and alternative hypotheses:

• H 0 (Null Hypothesis): μ ≤ 82
• H A (Alternative Hypothesis): μ > 82

To test this claim, the professor has 25 students use the new studying method and then take the exam. He collects the following data on the exam scores for this sample of students:

• t-test statistic: 3.6586
• one-tailed p-value: 0.0006

Since the p-value is less than .05, the professor rejects the null hypothesis.

He has sufficient evidence to conclude that the new studying method produces exam scores with an average score greater than 82.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing What is a Directional Hypothesis? When Do You Reject the Null Hypothesis?

Statistics vs. Probability: What’s the Difference?

One sample z-test calculator, related posts, how to normalize data between -1 and 1, how to interpret f-values in a two-way anova, how to create a vector of ones in..., vba: how to check if string contains another..., how to determine if a probability distribution is..., what is a symmetric histogram (definition & examples), how to find the mode of a histogram..., how to find quartiles in even and odd..., how to calculate sxy in statistics (with example), how to calculate sxx in statistics (with example).

AI, Analytics & Data Science: Towards Analytics Specialist

Unlocking the Power of One-Tailed Tests in Statistical Analysis: A Python-Based Approach

Article Outline:

1. Introduction to One-Tailed Testing - Definition and overview of one-tailed tests in statistical hypothesis testing. - Comparison with two-tailed tests: when and why to use one over the other. - Importance of one-tailed tests in research and data analysis.

2. Theoretical Foundations of One-Tailed Tests - Explanation of hypothesis testing: null hypothesis, alternative hypothesis, and significance levels. - Understanding the directionality in one-tailed tests: left-tailed and right-tailed tests. - Critical values and p-values in the context of one-tailed testing.

3. Determining When to Use One-Tailed Tests - Criteria for choosing a one-tailed test over a two-tailed test. - Common scenarios and examples where one-tailed tests are appropriate. - Potential pitfalls and misconceptions about one-tailed testing.

4. Conducting One-Tailed Tests with Python - Overview of Python libraries relevant to statistical testing (SciPy, Statsmodels). - Step-by-step guide to performing a one-tailed test in Python, including code snippets. - Preparing your dataset for analysis. - Choosing the appropriate statistical test (t-test, z-test) based on your data. - Adapting two-tailed test functions for one-tailed analysis. - Visualizing test results using Python libraries (Matplotlib, Seaborn).

5. Case Study: Applying a One-Tailed Test to a Public Dataset - Selection and description of a suitable publicly available dataset for analysis. - Formulation of research questions and hypotheses suitable for a one-tailed test. - Detailed walkthrough of data preprocessing, test execution, and result interpretation using Python. - Discussion of findings and their implications.

6. Challenges and Considerations in One-Tailed Testing - Ethical considerations and the risk of bias in choosing a one-tailed test. - Importance of data normality and sample size in one-tailed testing. - Adjusting significance levels and dealing with false positives.

7. Advanced Topics in One-Tailed Testing - Exploration of non-parametric one-tailed tests for data that do not meet parametric test assumptions. - The role of power analysis in one-tailed testing and how to conduct it with Python. - Bayesian approaches to one-tailed hypothesis testing.

8. Conclusion - Recap of the key points covered and the significance of one-tailed tests in statistical analysis. - The role of Python in facilitating robust and accessible statistical testing. - Encouragement for further exploration and application of one-tailed tests in research.

This article is designed to provide a comprehensive overview of one-tailed testing, blending theoretical understanding with practical application through Python. It aims to equip readers with the knowledge and tools necessary to confidently apply one-tailed tests in their research or data analysis projects.

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4 One-tailed vs two-tailed test

To gain a deeper understanding of how to conduct a hypothesis test, this section will delve into the concepts of one-tailed and two-tailed tests. These tests are vital tools in statistical hypothesis testing, and the decision of which test to employ depends on the research question and hypothesis under examination. It is crucial to give careful thought to the suitable type of test to ensure that the hypothesis is thoroughly tested and precise conclusions are derived from the data. This section will elaborate on this topic in greater detail.

To commence, complete the following activity pertaining to the formulation of null and alternative hypotheses. This exercise may be somewhat challenging, but it serves as an excellent introduction to upcoming discussions – don’t be concerned if you find it difficult!

Activity 3 Hypotheses setting

Read the following statements and then develop a null hypothesis and an alternative hypothesis.

‘It is believed that OU students need to set aside no longer than, on average, 15 hours to study an entire session of an OU course. However, a researcher believes that OU students spend longer studying an entire session of an OU course.’

H 0 : OU students spend, on average, no more than 15 hours studying an entire session of OU course.

H a : OU students spend, on average, more than 15 hours studying an entire session of OU course.

They can also be written as:

H 0 : µ ≤ 15 hours studies

H a : µ > 15 hours studies

µ is a symbol for a population mean. Remember, H 0 and H a are always opposites.

Did you identify any differences between the hypotheses you developed in Activity 1 and Activity 3? The set of hypotheses in Activity 1 has an equal (=) or not equal (≠) supposition (sign) in the statement. However, in Activity 3, the set of hypotheses has less than or equal to (≤) and greater than (>) supposition (sign) in the statement. This creates different conditions that lead to acceptance or rejection of the null hypothesis.

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• Knowledge Base

An Introduction to t Tests | Definitions, Formula and Examples

Published on January 31, 2020 by Rebecca Bevans . Revised on June 22, 2023.

A t test is a statistical test that is used to compare the means of two groups. It is often used in hypothesis testing to determine whether a process or treatment actually has an effect on the population of interest, or whether two groups are different from one another.

• The null hypothesis ( H 0 ) is that the true difference between these group means is zero.
• The alternate hypothesis ( H a ) is that the true difference is different from zero.

Table of contents

When to use a t test, what type of t test should i use, performing a t test, interpreting test results, presenting the results of a t test, other interesting articles, frequently asked questions about t tests.

A t test can only be used when comparing the means of two groups (a.k.a. pairwise comparison). If you want to compare more than two groups, or if you want to do multiple pairwise comparisons, use an   ANOVA test  or a post-hoc test.

The t test is a parametric test of difference, meaning that it makes the same assumptions about your data as other parametric tests. The t test assumes your data:

• are independent
• are (approximately) normally distributed
• have a similar amount of variance within each group being compared (a.k.a. homogeneity of variance)

If your data do not fit these assumptions, you can try a nonparametric alternative to the t test, such as the Wilcoxon Signed-Rank test for data with unequal variances .

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See an example

When choosing a t test, you will need to consider two things: whether the groups being compared come from a single population or two different populations, and whether you want to test the difference in a specific direction.

One-sample, two-sample, or paired t test?

• If the groups come from a single population (e.g., measuring before and after an experimental treatment), perform a paired t test . This is a within-subjects design .
• If the groups come from two different populations (e.g., two different species, or people from two separate cities), perform a two-sample t test (a.k.a. independent t test ). This is a between-subjects design .
• If there is one group being compared against a standard value (e.g., comparing the acidity of a liquid to a neutral pH of 7), perform a one-sample t test .

One-tailed or two-tailed t test?

• If you only care whether the two populations are different from one another, perform a two-tailed t test .
• If you want to know whether one population mean is greater than or less than the other, perform a one-tailed t test.
• Your observations come from two separate populations (separate species), so you perform a two-sample t test.
• You don’t care about the direction of the difference, only whether there is a difference, so you choose to use a two-tailed t test.

The t test estimates the true difference between two group means using the ratio of the difference in group means over the pooled standard error of both groups. You can calculate it manually using a formula, or use statistical analysis software.

T test formula

The formula for the two-sample t test (a.k.a. the Student’s t-test) is shown below.

In this formula, t is the t value, x 1 and x 2 are the means of the two groups being compared, s 2 is the pooled standard error of the two groups, and n 1 and n 2 are the number of observations in each of the groups.

A larger t value shows that the difference between group means is greater than the pooled standard error, indicating a more significant difference between the groups.

You can compare your calculated t value against the values in a critical value chart (e.g., Student’s t table) to determine whether your t value is greater than what would be expected by chance. If so, you can reject the null hypothesis and conclude that the two groups are in fact different.

T test function in statistical software

Most statistical software (R, SPSS, etc.) includes a t test function. This built-in function will take your raw data and calculate the t value. It will then compare it to the critical value, and calculate a p -value . This way you can quickly see whether your groups are statistically different.

In your comparison of flower petal lengths, you decide to perform your t test using R. The code looks like this:

Download the data set to practice by yourself.

Sample data set

If you perform the t test for your flower hypothesis in R, you will receive the following output:

The output provides:

• An explanation of what is being compared, called data in the output table.
• The t value : -33.719. Note that it’s negative; this is fine! In most cases, we only care about the absolute value of the difference, or the distance from 0. It doesn’t matter which direction.
• The degrees of freedom : 30.196. Degrees of freedom is related to your sample size, and shows how many ‘free’ data points are available in your test for making comparisons. The greater the degrees of freedom, the better your statistical test will work.
• The p value : 2.2e-16 (i.e. 2.2 with 15 zeros in front). This describes the probability that you would see a t value as large as this one by chance.
• A statement of the alternative hypothesis ( H a ). In this test, the H a is that the difference is not 0.
• The 95% confidence interval . This is the range of numbers within which the true difference in means will be 95% of the time. This can be changed from 95% if you want a larger or smaller interval, but 95% is very commonly used.
• The mean petal length for each group.

When reporting your t test results, the most important values to include are the t value , the p value , and the degrees of freedom for the test. These will communicate to your audience whether the difference between the two groups is statistically significant (a.k.a. that it is unlikely to have happened by chance).

You can also include the summary statistics for the groups being compared, namely the mean and standard deviation . In R, the code for calculating the mean and the standard deviation from the data looks like this:

flower.data %>% group_by(Species) %>% summarize(mean_length = mean(Petal.Length), sd_length = sd(Petal.Length))

In our example, you would report the results like this:

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.

• Chi square test of independence
• Statistical power
• Descriptive statistics
• Degrees of freedom
• Pearson correlation
• Null hypothesis

Methodology

• Double-blind study
• Case-control study
• Research ethics
• Data collection
• Hypothesis testing
• Structured interviews

Research bias

• Hawthorne effect
• Unconscious bias
• Recall bias
• Halo effect
• Self-serving bias
• Information bias

A t-test is a statistical test that compares the means of two samples . It is used in hypothesis testing , with a null hypothesis that the difference in group means is zero and an alternate hypothesis that the difference in group means is different from zero.

A t-test measures the difference in group means divided by the pooled standard error of the two group means.

In this way, it calculates a number (the t-value) illustrating the magnitude of the difference between the two group means being compared, and estimates the likelihood that this difference exists purely by chance (p-value).

Your choice of t-test depends on whether you are studying one group or two groups, and whether you care about the direction of the difference in group means.

If you are studying one group, use a paired t-test to compare the group mean over time or after an intervention, or use a one-sample t-test to compare the group mean to a standard value. If you are studying two groups, use a two-sample t-test .

If you want to know only whether a difference exists, use a two-tailed test . If you want to know if one group mean is greater or less than the other, use a left-tailed or right-tailed one-tailed test .

A one-sample t-test is used to compare a single population to a standard value (for example, to determine whether the average lifespan of a specific town is different from the country average).

A paired t-test is used to compare a single population before and after some experimental intervention or at two different points in time (for example, measuring student performance on a test before and after being taught the material).

A t-test should not be used to measure differences among more than two groups, because the error structure for a t-test will underestimate the actual error when many groups are being compared.

If you want to compare the means of several groups at once, it’s best to use another statistical test such as ANOVA or a post-hoc test.

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• Key Differences

Know the Differences & Comparisons

Difference Between One-tailed and Two-tailed Test

To test the hypothesis, test statistics is required, which follows a known distribution. In a test, there are two divisions of probability density curve, i.e. region of acceptance and region of rejection. the region of rejection is called as a critical region .

In the field of research and experiments, it pays to know the difference between one-tailed and two-tailed test, as they are quite commonly used in the process.

Content: One-tailed Test Vs Two-tailed Test

Comparison chart.

Definition of One-tailed Test

One-tailed test alludes to the significance test in which the region of rejection appears on one end of the sampling distribution. It represents that the estimated test parameter is greater or less than the critical value. When the sample tested falls in the region of rejection, i.e. either left or right side, as the case may be, it leads to the acceptance of alternative hypothesis rather than the null hypothesis. It is primarily applied in chi-square distribution; that ascertains the goodness of fit.

In this statistical hypothesis test, all the critical region, related to α , is placed in any one of the two tails. One-tailed test can be:

• Left-tailed test : When the population parameter is believed to be lower than the assumed one, the hypothesis test carried out is the left-tailed test.
• Right-tailed test : When the population parameter is supposed to be greater than the assumed one, the statistical test conducted is a right-tailed test.

Definition of Two-tailed Test

The two-tailed test is described as a hypothesis test, in which the region of rejection or say the critical area is on both the ends of the normal distribution. It determines whether the sample tested falls within or outside a certain range of values. Therefore, an alternative hypothesis is accepted in place of the null hypothesis, if the calculated value falls in either of the two tails of the probability distribution.

In this test, α is bifurcated into two equal parts, placing half on each side, i.e. it considers the possibility of both positive and negative effects. It is performed to see, whether the estimated parameter is either above or below the assumed parameter, so the extreme values, work as evidence against the null hypothesis.

Key Differences Between One-tailed and Two-tailed Test

The fundamental differences between one-tailed and two-tailed test, is explained below in points:

• One-tailed test, as the name suggest is the statistical hypothesis test, in which the alternative hypothesis has a single end. On the other hand, two-tailed test implies the hypothesis test; wherein the alternative hypothesis has dual ends.
• In the one-tailed test, the alternative hypothesis is represented directionally. Conversely, the two-tailed test is a non-directional hypothesis test.
• In a one-tailed test, the region of rejection is either on the left or right of the sampling distribution. On the contrary, the region of rejection is on both the sides of the sampling distribution.
• A one-tailed test is used to ascertain if there is any relationship between variables in a single direction, i.e. left or right. As against this, the two-tailed test is used to identify whether or not there is any relationship between variables in either direction.
• In a one-tailed test, the test parameter calculated is more or less than the critical value. Unlike, two-tailed test, the result obtained is within or outside critical value.
• When an alternative hypothesis has ‘≠’ sign, then a two-tailed test is performed. In contrast, when an alternative hypothesis has ‘> or <‘ sign, then one-tailed test is carried out.

To sum up, we can say that the basic difference between one-tailed and two-tailed test lies in the direction, i.e. in case the research hypothesis entails the direction of interrelation or difference, then one-tailed test is applied, but if the research hypothesis does not signify the direction of interaction or difference, we use two-tailed test.

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April 24, 2018 at 11:47 am

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July 23, 2020 at 11:48 pm

This website is very useful and easy to understand for Statistics methods and concepts.

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January 11, 2023 at 5:13 am

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