# Two-Tailed Test: Definition, Examples, and Importance in Statistics

![Two-Tailed Test](https://www.investopedia.com/thmb/XywcHohqDEGU0sdGxWt6jlu0t9M=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/two-tailed-test-4187540-1-50e779cf54484f16947c7abae4176c16.jpg)

*— Investopedia / Joules Garcia*

## What Is a Two-Tailed Test?

A two-tailed test in statistics evaluates whether a sample significantly differs from a population mean, either exceeding or falling short. It's a fundamental tool in null-hypothesis testing used across various fields like manufacturing quality control and financial analysis. When the sample mean falls within critical regions on either distribution tail, it signifies a departure from the population mean, prompting the rejection of the null hypothesis for the alternative. Understanding this test's principles and applications can enhance decision-making in data-driven settings.  

### Key Takeaways

*   A two-tailed test assesses whether a sample deviates significantly from the population mean by examining both ends of a distribution.
*   This type of test is crucial in hypothesis testing, as critical results can lead to the rejection of the null hypothesis in favor of the alternative.  
    
*   Two-tailed tests are applicable in diverse fields, such as manufacturing, to ensure samples meet specified quality standards.  
    
*   They differ from one-tailed tests, which evaluate if a sample mean is exclusively higher or lower than the population mean, but not both.  
    
*   An illustrative example involves testing a brokerage firm's claim on low fees, showing the practical use of a two-tailed test in real-world scenarios.

## Key Concepts of Two-Tailed Hypothesis Testing

Hypothesis testing is a [core idea](https://www.investopedia.com/terms/h/hypothesistesting.asp) in inferential statistics, used to assess the truth of a claim about a population parameter. A two-tailed test shows if the sample mean is significantly greater or less than the population mean. The test is named for examining the area under both tails of a normal distribution, but it also applies to non-normal distributions.

A two-tailed test is designed to examine both sides of a specified data range as designated by the [probability distribution](https://www.investopedia.com/terms/p/probabilitydistribution.asp) involved. The probability distribution should represent the likelihood of a specified outcome based on predetermined standards. This involves setting limits for the highest and lowest accepted values in the range. Any data point beyond these limits falls into the rejection range.

There is no inherent standard for the number of data points that must exist within the acceptance range. When precision is crucial, like in pharmaceutical drug creation, a rejection rate of 0.001% or less may be used. For less critical cases, like counting food items in a bag, a 5% rejection rate may suffice.

## Practical Applications of Two-Tailed Tests

A two-tailed test can be applied in production, like during candy packaging at a facility. If the target is 50 candies per bag, bags with fewer than 45 or more than 55 candies are rejected.

Random sampling can check if the packaging machinery is accurately calibrated for the expected output. A simple random sample selects a random portion of the population, giving each member an equal chance to be picked.

The packaging mechanism needs to average 50 candies per bag with an acceptable distribution to be considered accurate. The number of rejected bags should also stay within the acceptable error rate. Here, the null hypothesis would be that the mean is 50 while the alternate hypothesis would be that it is not 50.

If the z-score is in the rejection region, showing a large deviation from the mean, adjustments might be needed in the facility. Regularly using two-tailed tests can help keep production within limits long-term.  

> **Important**
>
> Be careful to note if a statistical test is one- or two-tailed as this will greatly influence a model's interpretation.

## Two-Tailed vs. One-Tailed Tests: Key Differences

A one-tailed test checks if the sample mean is only higher than the population mean. An example hypothesis is that "investment fund returns are at least x%. One-tailed tests could also be set up to show that the sample mean could be only less than the population mean. The key difference from a two-tailed test is that in a two-tailed test, the sample mean could be different from the population mean by being _either_ higher or lower than it.

If the sample being tested falls into the one-sided critical area, the alternative hypothesis will be accepted instead of the null hypothesis. A one-tailed test is also known as a directional hypothesis or directional test.

In contrast, a two-tailed test looks at both sides of a data range to see if a sample is above or below it.

## Example: How Two-Tailed Tests Work in Practice

As a hypothetical example, imagine that a new [stockbroker](https://www.investopedia.com/terms/s/stockbroker.asp), named XYZ, claims that their brokerage fees are lower than that of your current stockbroker, ABC) Data available from an independent research firm indicates that the mean and standard deviation of all ABC broker clients are $18 and $6, respectively.

A sample of 100 clients of ABC is taken, and brokerage charges are calculated with the new rates of XYZ broker. If the mean of the sample is $18.75 and the sample standard deviation is $6, can any inference be made about the difference in the average brokerage bill between ABC and XYZ broker?

*   H0: Null Hypothesis: mean = 18
*   H1: Alternative Hypothesis: mean <> 18 (This is what we want to prove.)
*   Rejection region: Z <= - Z2.5 and Z>=Z2.5 (assuming 5% significance level, split 2.5 each on either side).
*   Z = (sample mean – mean) / (std-dev / sqrt (no. of samples)) = (18.75 – 18) / (6/(sqrt(100)) = 1.25

This calculated Z value falls between the two limits defined by: - Z2.5 \= -1.96 and Z2.5 \= 1.96.

![Image](https://www.investopedia.com/thmb/TDKV_REnO8ek-F-_Hsnhq_gm0XI=/1500x0/filters:no_upscale():max_bytes(150000):strip_icc()/dotdash_Final_Two_Tailed_Test_Sep_2020-01-b86f83d6c5f64f1a8e091d1cb0cd5f67.jpg)

**

This concludes that there is insufficient evidence to infer that there is any difference between the rates of your existing broker and the new broker. Therefore, the null hypothesis cannot be rejected. Alternatively, the p-value = P(Z< -1.25)+P(Z >1.25) = 2 \* 0.1056 = 0.2112 = 21.12%, which is greater than 0.05 or 5%, leads to the same conclusion.

## How Is a Two-Tailed Test Designed?

A two-tailed test is designed to determine whether a claim is true or not given a population parameter. It examines both sides of a specified data range as designated by the probability distribution involved. As such, the probability distribution should represent the likelihood of a specified outcome based on predetermined standards.

## What Is the Difference Between a Two-Tailed and One-Tailed Test?

A two-tailed hypothesis test is designed to show whether the sample mean is significantly greater than _or_ significantly less than the mean of a population. The two-tailed test gets its name from testing the area under both tails (sides) of a normal distribution. A one-tailed hypothesis test, on the other hand, is set up to show only one test; that the sample mean would be higher than the population mean, or, in a separate test, that the sample mean would be lower than the population mean.

## What Is a Z-score?

A Z-score numerically describes a value's relationship to the mean of a group of values and is measured in terms of the number of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score whereas Z-scores of 1.0 and -1.0 would indicate values one standard deviation above or below the mean. In most large data sets, 99% of values have a Z-score between -3 and 3, meaning they lie within three standard deviations above and below the mean.

## The Bottom Line

A two-tailed test plays a crucial role in hypothesis testing by determining if a sample mean significantly differs from a population mean in either direction, assessing both tails of a distribution. This statistical tool helps replace the null hypothesis with the alternative when results fall in the critical regions, ensuring accuracy and reliability in various fields such as manufacturing quality control and scientific research. Understanding the distinction between two-tailed and one-tailed tests is essential, as it greatly impacts the interpretation of statistical models.