Sometimes tests of significance or hypothesis tests are referred to by the names one-tailed or two-tailed tests. These names are quite helpful in describing the hypothesis test, but it can be unclear to beginners to the subject where these names come from. What is a one-tailed test?

One-Tailed Tests

The number of tails in a test refers to the shape of the probability distribution being used. Hypothesis tests involve calculating the likelihood of observing by chance alone a value at least as extreme as the value of our test statistic.


Equivalently, the p-value of a test is the area under the curve of the distribution at these extreme values. Since probability distributions must eventually get closer to the x-axis, these extreme values look like a tail on the graph of the probability distribution.

The presence of a one-tailed test can be easily identified by the form of the statement of the alternative hypothesis. If we have a strict inequality in our alternative hypothesis, then we should expect to use a one-tailed test.

Since both “less than” or “greater than” is possible, there are further considerations with a one-tailed hypothesis test.

Further Distinction

One-tailed tests can be classified with greater specificity. Since two types of strict inequalities give rise to a one-tailed test, we can have separate names for each of these types of tests. The terminology “upper-tailed” and “lower-tailed” can be used. These names give further indication of the position in the distribution under consideration.

Upper-tailed tests refer to the extreme right of the distribution.

This corresponds to an alternate hypothesis of a population parameter being greater than a particular value. The p-value of our test corresponds to the area to the right of a point in the distribution.

Lower-tailed tests refer to the extreme left of the distribution. This corresponds to an alternate hypothesis of a population parameter being less than a particular value. The p-value of our test is equivalent to the area to the left of a point in the distribution.

Examples of One-Tailed Tests

One-tailed tests are used throughout statistics. A few of the most common instances of this type of test include the following:

• Tests about the population mean with an alternate hypothesis of “less than” or “greater than.” Both upper-tailed and lower-tailed tests are possible as these tests use the symmetric distributions of the standard normal distribution and Student’s t-distribution.
• Tests about the difference of two independent population means with an alternative hypothesis of the difference of these means being less than zero or the difference of the means is greater than zero.
• Tests about a population proportion in which the alternative hypothesis contains a strict inequality.
• Tests involving the chi-square distribution such as a goodness of fit test are all one-tailed tests. Due to the distribution being defined for only nonnegative values, these tests are all upper-tailed tests.
• Tests involving the F-distribution such as one factor analysis of variance (ANOVA) are all one-tailed tests. As with the chi-square distribution, due to the distribution being defined for only nonnegative values, these tests are all upper-tailed tests.