1- vs 2-Tailed Tests

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1- and 2-Tailed Tests

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On this page, you will learn the difference between one- and two-tailed tests.

One-tailed tests

In the previous pages, you learned how to perform define the hypothesis for a statistical test, then to perform a t-test to compare means. In the example t-test we performed, we defined an alternate hypothesis to test whether one mean was greater than the other: μ > μ0.

In this situation, we tested whether one mean was higher than the other. We were not interested in whether the first mean was lower than the other, only if it was higher. So we were only interested in one side of the probability distribution, which is shown in the image below:


In this distribution, the shaded region shows the area represented by the null hypothesis, H0: μ = μ0. This actually implies μμ0, since the only unshaded region in the image shows μ > μ0. Because we were only interested in one side of the distribution, or one "tail", this type of test is called a one-sided or a one-tailed test. When you are using tables for probability distributions, you should make sure whether they are for one-tailed or two-tailed tests. Depending on which they are for, you need to know how to switch to the one you need. This is all explained below.

A one-tailed test uses an alternate hypothesis that states either H1: μ > μ0 OR H1: μ < μ0, but not both. If you want to test both, using the alternate hypothesis H1: μμ0, then you need to use a two-tailed test.

Two-tailed tests

We would use a two-tailed test to see if two means are different from each other (ie from different populations), or from the same population. As an example, let's assume that we want to check if the pH of a stream has changed significantly in the past year. A water sample from the stream was analyzed using a pH electrode, where six samples were taken. It was found that the mean pH reading was 6.5 with standard deviation sold = 0.2. A year later, six more samples were analyzed, and the mean pH of these readings was 6.8 with standard deviation sold = 0.1.

Example 1

We could use a one-tailed test, to see if the stream has a higher pH than one year ago, for which we would use the alternate hypothesis HA: μprev < μcurrent. However, we may want a more rigorous test, for the hypothesis that HA: μprevμcurrent. This would mean that both HA: μprev < μcurrent and HA: μprev > μcurrent were satisfied, and we could be sure that there is a significant difference between the means. The probability distribution for a 90% confidence level, two-tailed test looks like this:


Continuing the example, we define the null hypothesis H0: μprev = μcurrent, and the alternate hypothesis HA: μprevμcurrent. The d.o.f. for a two sample mean t-test is ν = 7.35 ≈ 7, since the d.o.f. must be a whole number. The t-value for the two sample test is

t = (6.8-6.5)/sqrt((0.1)^2/6 + (0.2)^2/6) = 3.29

If we consult a two-tailed t-test table, for a 95% confidence limit, we find that t7,95% = 2.36. Since tcalc > t7,95%, we reject the null hypothesis, accept the alternate hypothesis that μprevμcurrent, and can say that the means are significantly different.

Using Tables for One- and Two-Tailed Tests

Some tables of critical t-values only give you the values for either a one- or two-tailed test, but not both. Because of this, you will need to know how to use one-tailed tables for two-tailed tests, and vice versa. The conversion is actually quite simple:

Table you have Operation To get ...
One-tailed Divide P by 2 Two-tailed test for P/2
Two-tailed Multiply P by 2 One-tailed test for 2P

For example, assume you have a table to a one-tail test at the 98% confidence level and want to perform a two-tailed test. For the 98% confidence level, P = 0.02. Divide P by 2 to get 0.01, which is a 99% confidence level. So you would compare tcalc to the value from the 98% one-tailed table, and it would be equivalent to a two-tailed test at the 99% confidence level.


© Dr. David Stone (dstone at chem.utoronto.ca) & Jon Ellis (jon.ellis at utoronto.ca) , August 2006