Statistics Tutorial

Paired t-test

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Descriptive StatisticsHistograms | Central Tendency | Standard Deviation | Confidence Intervals
Comparing Two SamplesSamples and Populations | Choosing a T-Test | Paired T-Test | P-Values and T-Tables
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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.


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


Use the calculator to the right to work through the formula above and work out the t-value for your data. Keep your subtotals in the calculator memory so that you do not loose accuracy with rounding errors. Only round your final answer.

There are 30 data points, so n=30.

  1. Click here to calculate the sum of the differences and the sum of the squared differences between the paired observations. Make sure your browser allows pop-ups so that you can see this data.
  2. Multiply n by the sum of the squared differences
  3. Subtract the sum of the differences squared from the current answer.
  4. Divide the current answer by n-1
  5. Now find the square root of the current answer (Sqrt button) and put the answer into memory (M= button)
  6. Now Enter the sum of the differences value and divide by the value stored in memory (MR button recalls from memory).
  7. Finally, use the RD button to round your answer to 3 decimal places and enter the answer below.
Enter your value for t here:   Help

Note that it is perfectly okay if you get a negative value for t. If you do, remember to put the minus sign (-) before the number you enter above.

( You need to enable Java to see this applet. )
Choosing a T-Test | P-Values and T-Tables