# Understanding One-Tailed Tests: Definition, Example, and Significance

![One-Tailed Test](https://www.investopedia.com/thmb/8e1eQdOcUywgHlK8Vi5QTPoPPuw=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/One-TailedTest-2f82ee3d082f4dd3bd57c14ce29315b8.jpg)

*— Investopedia / Xiaojie Liu*

One-tailed tests in statistics focus on one direction in the distribution to determine if a sample mean is significantly greater or less than a population mean. This approach is especially useful in finance when testing hypotheses, such as an investment outperforming the market. If the data falls in the critical region, statisticians reject the null hypothesis in favor of the alternative.

### Key Takeaways

*   A one-tailed test in statistics is used to determine if a sample mean is significantly higher or lower than a population mean.
*   This test focuses on the possibility of a relationship in only one direction, either greater than or less than, but not both.
*   Analysts utilize the one-tailed test to validate investment hypotheses by establishing null and alternative hypotheses.
*   The significance level in a one-tailed test is often set at 1%, 5%, or 10% to measure the probability of rejecting a true null hypothesis.
*   A one-tailed test is chosen when it is unnecessary to consider outcomes in the opposite direction of interest.

## What Is a One-Tailed Test?

[Hypothesis testing](https://www.investopedia.com/terms/h/hypothesistesting.asp) is a key concept in inferential statistics, used to check if a claim about a population is true. 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](https://www.investopedia.com/terms/t/two-tailed-test.asp). A one-tailed test checks whether a sample mean is higher or lower than the population mean. The one-tailed test gets its name from testing the area under one of the tails (sides) of a [normal distribution](https://www.investopedia.com/terms/n/normaldistribution.asp), although the test can be used in other non-normal distributions.

Before conducting a one-tailed test, establish null and alternative hypotheses. The [null hypothesis](https://www.investopedia.com/terms/n/null_hypothesis.asp) is the claim to reject, while the alternative hypothesis is accepted if the null is rejected.

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

## Real-World Application of One-Tailed Tests

Suppose an analyst wants to show that a [portfolio manager](https://www.investopedia.com/terms/p/portfoliomanager.asp) outperformed the [S&P 500 index](https://www.investopedia.com/terms/s/sp500.asp) by 16.91% in a year. They may set up the null (H0) and alternative (Ha) hypotheses as:

H0: μ ≤ 16.91

Ha: μ > 16.91

The null hypothesis is the measurement that the analyst hopes to reject. The alternative hypothesis claims that the portfolio manager did better than the S&P 500. If the test rejects the null hypothesis, the alternative hypothesis is supported. If not, further analysis of the portfolio manager’s performance may be needed.

The region of rejection is on only one side of the sampling distribution in a one-tailed test. To compare the portfolio's [return on investment](https://www.investopedia.com/terms/r/returnoninvestment.asp) to the market, the analyst performs an upper-tailed test, focusing on extreme values in the right side of the 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.

## Assessing Statistical Significance in One-Tailed Tests

To determine how significant the difference in returns is, a [significance level](https://www.investopedia.com/terms/s/statistically_significant.asp) 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](https://www.investopedia.com/terms/p/p-value.asp), the stronger the evidence that the null hypothesis is false.

If the p-value is less than 5%, the difference is 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 H0 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.

In a one-tailed test, the analyst checks for a relationship in only one direction, ignoring the other. 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.

## The Bottom Line

A one-tailed test is a critical tool in hypothesis testing when the focus is on assessing a potential increase or decrease relative to a population parameter. It allows analysts to test investment or portfolio hypotheses with a focus on one direction, disregarding opposite outcomes. Key steps include setting up null and alternative hypotheses and determining a p-value for significance. Utilizing a one-tailed test can lead to more focused and relevant conclusions, but it should only be applied when opposite outcomes are not critical to the analysis.

> **Article Sources**
>
> 1.  University of Southern California. "[FAQ: What Are the Differences Between One-Tailed and Two-Tailed Tests?](https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faq-what-are-the-differences-between-one-tailed-and-two-tailed-tests/)"