Two-sample t-tests are statistical tests used to compare the means of two populations. Also known as Student’s t-tests, their results are used to determine if there is a significant difference between the mean of two samples that is unlikely to be due to sampling error or random chance.

Student’s t-tests are further broken down into two categories: paired t-tests and unpaired t-tests. These statistical tests are commonly used in research in the fields of biology, business, and psychology.

This article will explain when it is appropriate to use a paired t-test versus an unpaired t-test, as well as the hypothesis and assumptions of each.

## What is a paired t-test?

A paired t-test (also known as a dependent or correlated t-test) is a statistical test that compares the averages/means and standard deviations of two related groups to determine if there is a significant difference between the two groups.

● A significant difference occurs when the differences between groups are unlikely to be due to sampling error or chance.

● The groups can be related by being the same group of people, the same item, or being subjected to the same conditions.

Paired t-tests are considered more powerful than unpaired t-tests because using the same participants or item eliminates variation between the samples that could be caused by anything other than what’s being tested.

### What are the hypotheses of a paired t-test?

There are two possible hypotheses in a paired t-test.

● The **null hypothesis** (H0) states that there is no significant difference between the means of the two groups.

● The **alternative hypothesis** (H1) states that there is a significant difference between the two population means, and that this difference is unlikely to be caused by sampling error or chance.

### What are the assumptions of a paired t-test?

● The dependent variable is normally distributed

● The observations are sampled independently

● The dependent variable is measured on an incremental level, such as ratios or intervals.

● The independent variables must consist of two related groups or matched pairs.

## When to use a paired t-test?

Paired t-tests are used when the same item or group is tested twice, which is known as a repeated measures t-test. Some examples of instances for which a paired t-test is appropriate include:

● The before and after effect of a pharmaceutical treatment on the same group of people.

● Body temperature using two different thermometers on the same group of participants.

● Standardized test results of a group of students before and after a study prep course.

## What is an unpaired t-test?

An unpaired t-test (also known as an independent t-test) is a statistical procedure that compares the averages/means of two independent or unrelated groups to determine if there is a significant difference between the two.

### What are the hypotheses of an unpaired t-test?

The hypotheses of an unpaired t-test are the same as those for a paired t-test. The two hypotheses are:

● The null hypothesis (H0) states that there is no significant difference between the means of the two groups.

● The alternative hypothesis (H1) states that there is a significant difference between the two population means, and that this difference is unlikely to be caused by sampling error or chance.

### What are the assumptions of an unpaired t-test?

● The dependent variable is normally distributed

● The observations are sampled independently

● The dependent variable is measured on an incremental level, such as ratios or intervals.

● The variance of data is the same between groups, meaning that they have the same standard deviation

● The independent variables must consist of two independent groups.

## When to use an unpaired t-test?

An unpaired t-test is used to compare the mean between two independent groups. You use an unpaired t-test when you are comparing two separate groups with equal variance.

Examples of appropriate instances during which to use an unpaired t-test:

● Research, such as a pharmaceutical study or other treatment plan, where ½ of the subjects are assigned to the treatment group and ½ of the subjects are randomly assigned to the control group.

● Research during which there are two independent groups, such as women and men, that examines whether the average bone density is significantly different between the two groups.

● Comparing the average commuting distance traveled by New York City and San Francisco residents using 1,000 randomly selected participants from each city.

In the case of unequal variances, a Welch’s test should be used.

## Paired vs unpaired t-test

The key differences between a paired and unpaired t-test are summarized below.

- A paired t-test is designed to compare the means of the same group or item under two separate scenarios. An unpaired t-test compares the means of two independent or unrelated groups.
- In an unpaired t-test, the variance between groups is assumed to be equal. In a paired t-test, the variance is not assumed to be equal.