Choosing whether to perform a one-tailed or a two-tailed hypothesis test is one of the methodology decisions you might need to make for your statistical analysis. This choice can have critical implications for the types of effects it can detect, the statistical power of the test, and potential errors.
In this post, you’ll learn about the differences between one-tailed and two-tailed hypothesis tests and their advantages and disadvantages. I include examples of both types of statistical tests. In my next post, I cover the decision between one and two-tailed tests in more detail.
What Are Tails in a Hypothesis Test?
First, we need to cover some background material to understand the tails in a test. Typically, hypothesis tests take all of the sample data and convert it to a single value, which is known as a test statistic. You’re probably already familiar with some test statistics. For example, t-tests calculate t-values. F-tests, such as ANOVA, generate F-values. The chi-square test of independence and some distribution tests produce chi-square values. All of these values are test statistics.
These test statistics follow a sampling distribution. Probability distribution plots display the probabilities of obtaining test statistic values when the null hypothesis is correct. On a probability distribution plot, the portion of the shaded area under the curve represents the probability that a value will fall within that range.
The graph below displays a sampling distribution for t-values. The two shaded regions cover the two-tails of the distribution.
Keep in mind that this t-distribution assumes that the null hypothesis is correct for the population. Consequently, the peak (most likely value) of the distribution occurs at t=0, which represents the null hypothesis in a t-test. Typically, the null hypothesis states that there is no effect. As t-values move further away from zero, it represents larger effect sizes. When the null hypothesis is true for the population, obtaining samples that exhibit a large apparent effect becomes less likely, which is why the probabilities taper off for t-values further from zero.
Critical Regions in a Hypothesis Test
In hypothesis tests, critical regions are ranges of the distributions where the values represent statistically significant results. Analysts define the size and location of the critical regions by specifying both the significance level (alpha) and whether the test is one-tailed or two-tailed.
Consider the following two facts:
- The significance level is the probability of rejecting a null hypothesis that is correct.
- The sampling distribution for a test statistic assumes that the null hypothesis is correct.
Consequently, to represent the critical regions on the distribution for a test statistic, you merely shade the appropriate percentage of the distribution. For the common significance level of 0.05, you shade 5% of the distribution.
Related post: Significance Levels and P-values
Two-Tailed Hypothesis Tests
Two-tailed hypothesis tests are also known as nondirectional and two-sided tests because you can test for effects in both directions. When you perform a two-tailed test, you split the significance level percentage between both tails of the distribution. In the example below, I use an alpha of 5% and the distribution has two shaded regions of 2.5% (2 * 2.5% = 5%).
When a test statistic falls in either critical region, your sample data are sufficiently incompatible with the null hypothesis that you can reject it for the population.
In a two-tailed test, the generic null and alternative hypotheses are the following:
- Null: The effect equals zero.
- Alternative: The effect does not equal zero.
The specifics of the hypotheses depend on the type of test you perform because you might be assessing means, proportions, or rates.
Example of a two-tailed 1-sample t-test
Suppose we perform a two-sided 1-sample t-test where we compare the mean strength (4.1) of parts from a supplier to a target value (5). We use a two-tailed test because we care whether the mean is greater than or less than the target value.
To interpret the results, simply compare the p-value to your significance level. If the p-value is less than the significance level, you know that the test statistic fell into one of the critical regions, but which one? Just look at the estimated effect. In the output below, the t-value is negative, so we know that the test statistic fell in the critical region in the left tail of the distribution. This occurs when the mean is less than the target value, and now we know that this difference is statistically significant.
We can conclude that the population mean for part strength is less than the target value. However, the test had the capability to detect a positive difference as well. You can also assess the confidence interval. With a two-tailed hypothesis test, you’ll obtain a two-sided confidence interval. The confidence interval tells us that population mean is likely to fall between 3.372 and 4.828.
Advantages of two-tailed hypothesis tests
You can detect both positive and negative effects. Two-tailed tests are standard in scientific research where discovering any type of effect is usually of interest to researchers.
One-Tailed Hypothesis Tests
One-tailed hypothesis tests are also known as directional and one-sided tests because you can test for effects in only one direction. When you perform a one-tailed test, the entire significance level percentage goes into the extreme end of one tail of the distribution.
In the examples below, I use an alpha of 5%. Each distribution has one shaded region of 5%. When you perform a one-tailed test, you must determine whether the critical region is in the left tail or the right tail. The test can detect an effect only in the direction that has the critical region. It has absolutely no capacity to detect an effect in the other direction.
In a one-tailed test, you have two options for the null and alternative hypotheses, which corresponds to where you place the critical region.
You can choose either of the following sets of generic hypotheses:
- Null: The effect is less than or equal to zero.
- Alternative: The effect is greater than zero.
- Null: The effect is greater than or equal to zero.
- Alternative: The effect is less than zero.
Again, the specifics of the hypotheses depend on the type of test you perform.
Notice how for both possible null hypotheses the tests can’t distinguish between zero and an effect in a particular direction. For example, in the example directly above, the null combines “the effect is greater than or equal to zero” into a single category. That test can’t differentiate between zero and greater than zero.
Example of a one-tailed 1-sample t-test
Suppose we perform a one-tailed 1-sample t-test. We’ll use a similar scenario as before where we compare the mean strength of parts from a supplier (102) to a target value (100). Imagine that we are considering a new parts supplier. We will use them only if the mean strength of their parts is greater than our target value. There is no need for us to differentiate between whether their parts are equally strong or less strong than the target value—either way we’d just stick with our current supplier.
Consequently, we’ll choose the alternative hypothesis that states the mean difference is greater than zero (Population mean – Target value > 0). The null hypothesis states that the difference between the population mean and target value is less than or equal to zero.
To interpret the results, compare the p-value to your significance level. If the p-value is less than the significance level, you know that the test statistic fell into the critical region. For this study, the statistically significant result supports the notion that the population mean is greater than the target value of 100.
Confidence intervals for a one-tailed test are similarly one-sided. You’ll obtain either an upper bound or a lower bound. In this case, we obtain a lower bound, which indicates that the population mean is likely to be greater than or equal to 100.631. There is no upper limit to this range. This one-sided lower-bound matches our goal of determining only whether the new parts are stronger than our target value. We only need to know that the parts are stronger.
This test is unable to detect a negative difference even when the sample mean represents a very negative effect.
Advantages and disadvantages of one-tailed hypothesis tests
One-tailed tests have more statistical power to detect an effect in one direction than a two-tailed test with the same design and significance level. One-tailed tests occur most frequently for studies where one of the following is true:
- Effects can exist in only one direction.
- Effects can exist in both directions but the researchers only care about an effect in one direction. There is no drawback to failing to detect an effect in the other direction. (Not recommended.)
The disadvantage of one-tailed tests is that is has no statistical power to detect an effect in the other direction.
This post explains the differences between one-tailed and two-tailed statistical hypothesis tests. How these forms of hypothesis tests function is clear and based on mathematics. However, there is some debate about when you can use one-tailed tests. My next post explores this decision in much more depth and explains the different schools of thought, as well as my own opinion on the matter—When Can I Use One-Tailed Hypothesis Tests.