Enter the mean of the differences, hypothesized mean difference, standard deviation of the differences, and the number of pairs into the calculator to determine the test statistic.

Paired Difference Test Formula

The following formula is used to calculate the Paired Difference Test.

t = (D̄ - μD) / (SD / √n)


  • t is the test statistic D̄ is the mean of the differences
  • μD is the hypothesized mean difference (usually 0)
  • SD is the standard deviation of the differences
  • n is the number of pairs

To calculate the test statistic, subtract the hypothesized mean difference from the mean of the differences. Divide this result by the standard deviation of the differences divided by the square root of the number of pairs.

What is a Paired Difference Test?

A Paired Difference Test is a statistical procedure used to determine whether the mean difference between two sets of observations is zero. It is often used in before-and-after situations where the same subjects are measured twice, under different conditions. The test pairs each observation in one set with an observation in the other set, calculates the differences within each pair, and analyzes these differences. The test assumes that the differences in the entire population of pairs follow a normal distribution.

How to Calculate Paired Difference Test?

The following steps outline how to calculate a Paired Difference Test using the formula: t = (D̄ – μD) / (SD / √n).

  1. First, determine the mean of the differences (D̄).
  2. Next, determine the hypothesized mean difference (μD). This is usually 0 unless stated otherwise.
  3. Next, determine the standard deviation of the differences (SD).
  4. Next, determine the number of pairs (n).
  5. Finally, calculate the test statistic (t) using the formula: t = (D̄ – μD) / (SD / √n).
  6. After inserting the variables and calculating the result, interpret the test statistic to make a conclusion about the paired difference test.

Example Problem : 

Use the following variables as an example problem to test your knowledge.

mean of the differences (D̄) = 2.5

hypothesized mean difference (μD) = 0

standard deviation of the differences (SD) = 1.2

number of pairs (n) = 20