## Paired t-test

 Getting Started General Instructions | Introduction to Your Study | Experimental Design | Stating a Hypothesis Descriptive Statistics Histograms | Central Tendency | Standard Deviation | Confidence Intervals Comparing Two Samples Samples and Populations | Choosing a T-Test | Paired T-Test | P-Values and T-Tables Important Concepts The Normal Distribution | Z Scores | Probability Distributions Levels You are currently on Paired T-Test at level 2. Level 1 | Level 2 | Level 3 Next Topic Choosing a T-Test | P-Values and T-Tables

### Explanation

The paired t-test is calculated to take into account the fact that pairs of subjects (one from each condition) go together. It is based on the differences between the values of each pair - that is one subtracted from the other. In the formula for a paired t-test, this difference is notated as d.

When you look at the formula for the paired t-test below, you will see that it uses just d and n (the number of values in the data), and nothing else. The way these two values effect the value of t are as follows:

• As the average of the differences gets bigger, t gets bigger;
• As the variation in the differences gets bigger, t gets smaller;
• As the number of values gets bigger, t gets bigger.
There is another way of writing the paired t-test formula, that you might see in a book. It makes the above points clearer, but is not so easy to use to calculate a t-value from data. It is shown in the help topic below if you are interested in seeing it.

### Exploration

Here is the formula for a paired t-test. Hover over any part to see that part explained.

The top of the formula is the sum of the differences (i.e. the sum of d). The bottom of the formula reads as:

The square root of the following: n times the sum of the differences squared minus the sum of the squared differences, all over n-1.

• The sum of the squared differences: ∑d2 means take each difference in turn, square it, and add up all those squared numbers.
• The sum of the differences squared: (∑d)2means add up all the differences and square the result.

Brackets around something in a formula mean (do this first), so (∑d)2 means add up all the differences first, then square the result.