T-Test Essentials: Definition, Formula and Calculation

Paired T-Test Formula

This article describes the paired t-test formula, which is used to compare the means of two related groups or samples. The paired t-test formula is also referred as:

  • dependent t test formula,
  • paired sample t test formula,
  • paired samples t test formula,
  • formula for paired t test,
  • paired t test equation and
  • dependent t test equation

The procedure of the paired t-test analysis is as follow:

  1. Calculate the difference (\(d\)) between each pair of value
  2. Compute the mean (\(m\)) and the standard deviation (\(s\)) of \(d\)
  3. Compare the average difference to 0. If there is any significant difference between the two pairs of samples, then the mean of d (\(m\)) is expected to be far from 0.


Related Book

Practical Statistics in R II - Comparing Groups: Numerical Variables


The paired t-test statistics value can be calculated using the following formula:

t = \frac{m}{s/\sqrt{n}}


  • m is the mean differences
  • n is the sample size (i.e., size of d).
  • s is the standard deviation of d

We can compute the p-value corresponding to the absolute value of the t-test statistics (|t|) for the degrees of freedom (df): \(df = n - 1\).

If the p-value is inferior or equal to 0.05, we can conclude that the difference between the two paired samples are significantly different.

Related article

T-test in R

Version: Français

Independent T-Test Formula (Prev Lesson)
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