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What is the difference between a two-tailed and a one-tailed test?

2017-10-13

A statistical test is based on two competing hypotheses: the null hypothesis H0 and the alternative hypothesis Ha.

The type of alternative hypothesis Ha defines if a test is one-tailed or two-tailed. 

Two-tailed tests

A Two-tailed test is associated to an alternative hypotheses for which the sign of the potential difference is unknown. For example, suppose we wish to compare the averages of two samples A and B. Before setting up the experiment and running the test, we expect that if a difference between the two averages is highlighted, we do not really know whether A would be higher than B or the opposite. This drives us to choose a two-tailed test, associated to the following alternative hypothesis: Ha: average(A) ≠ average(B). Two-tailed tests are by far the most commonly used tests.

One-tailed tests

A One-tailed test is associated to an alternative hypothesis for which the sign of the potential difference is known before running the experiment and the test. In the example described above, the alternative hypothesis related to a one-tailed test could be written as follows: average(A) < average(B) or average(A) > average(B), depending on the expected direction of the difference.

In most of the XLSTAT statistical test dialog boxes, the user is able to choose between two-tailed or one-tailed tests (Options tab, usually).

Going further: why are we talking about tails?

Statistical tests often imply the calculation of a specific number called a statistic. This number has a theoretical distribution under the null hypothesis. The distribution is bell-shaped in many cases. Have a look at the image below. Tails are the extremes of the bell.
In order to obtain the p-value, the statistic computed out of the data is compared to the distribution under the null hypothesis. The further the statistic from the center of the distribution (i.e. the closer it is to a tail), the more extreme the data is compared to data generated under the null hypothesis and thus the lower the p-value. Before gathering the data, if we don’t know whether the computed statistic will be at the right tail or at the left tail under the alternative hypothesis, the two tailed are considered. We thus work with a two-tailed hypothesis.
Two-tailed test, distribution
If the computed statistic is in one of the two grey areas, the p-value will be under the alpha threshold and thus the null hypothesis rejected.

Besides, if we expect a specific tail under the alternative hypothesis, it is possible to choose a one-tailed hypothesis.
One-tailed test, distribution
If the computed statistic is under the grey area, the p-value will be under the alpha threshold and thus the null hypothesis rejected.

The case of a Student’s t-test for comparing two averages

In the case of a Student’s t-test for comparing two averages, the computed statistic is called t. Computed t’s located at the right of the center of the distribution depict a positive difference between the two averages. t’s located to the left of the center of the distribution reflect a negative difference between the two averages.
Prior to collecting the data, we’re often unsure whether the potential difference between the two averages will be positive or negative. In other words, we don’t know if the computed t-statistic will be at the right tail or at the left tail of the bell under the alternative hypothesis. This is when a two-tailed hypothesis is appropriate.
When the sign of the expected difference between the two averages is known, we’re able to only concentrate on the expected tail under the alternative hypothesis. We thus choose a one-tailed hypothesis with the desired tail or direction.

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