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* > *μ*_{0}.

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, *H*_{0}:
*μ* = *μ*_{0}. This actually implies *μ* ≤ *μ*_{0}, since
the only unshaded region in the image shows *μ* > *μ*_{0}. 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 *H*_{1}:
*μ* > *μ*_{0} *OR**H*_{1}:
*μ* < *μ*_{0}, but not both. If you want to test both, using the alternate hypothesis
*H*_{1}: *μ* ≠ *μ*_{0}, 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 *s*_{old} = 0.2.
A year later, six more samples were analyzed, and the mean pH of these readings was 6.8 with standard deviation
*s*_{old} = 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 *H*_{A}: *μ*_{prev} < *μ*_{current}.
However, we may want a more rigorous test, for the hypothesis that *H*_{A}: *μ*_{prev}
≠ *μ*_{current}. This would mean that both *H*_{A}: *μ*_{prev}
< *μ*_{current} and *H*_{A}: *μ*_{prev} > *μ*_{current}
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 *H*_{0}: *μ*_{prev}
= *μ*_{current}, and the alternate hypothesis *H*_{A}: *μ*_{prev}
≠ *μ*_{current}. 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

If we consult a two-tailed *t*-test table, for a 95% confidence limit, we find that *t*_{7,95%}
= 2.36. Since *t*_{calc} > *t*_{7,95%}, we reject the null hypothesis, accept the alternate
hypothesis that *μ*_{prev} ≠ *μ*_{current}, 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 *t*_{calc} to the value from the 98% one-tailed table, and it would be equivalent
to a two-tailed test at the 99% confidence level.