# 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.

Contents:

#### Related Book

Practical Statistics in R II - Comparing Groups: Numerical Variables

## Formula

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

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

where,

• 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 