T-Test Essentials: Definition, Formula and Calculation

Independent T-Test Formula

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

  • unpaired t-test formula,
  • independent samples t-test formula,
  • two sample t-test formula,
  • 2 sample t-test formula and
  • two sample t-test equation

The independent samples t-test comes in two different forms:

  • the standard Student’s t-test, which assumes that the variance of the two groups are equal.
  • the Welch’s t-test, which is less restrictive compared to the original Student’s test. This is the test where you do not assume that the variance is the same in the two groups, which results in the fractional degrees of freedom.

In this article, you will learn the Student t-test formula and the Weltch t-test formula.


Related Book

Practical Statistics in R II - Comparing Groups: Numerical Variables


  1. Classical two independent samples t-test (Student t-test). If the variance of the two groups are equivalent (homoscedasticity), the t-test value, comparing the two samples (\(A\) and \(B\)), can be calculated as follow.

t = \frac{m_A - m_B}{\sqrt{ \frac{S^2}{n_A} + \frac{S^2}{n_B} }}


  • \(m_A\) and \(m_B\) represent the mean value of the group A and B, respectively.
  • \(n_A\) and \(n_B\) represent the sizes of the group A and B, respectively.
  • \(S^2\) is an estimator of the pooled variance of the two groups. It can be calculated as follow :

S^2 = \frac{\sum{(x-m_A)^2}+\sum{(x-m_B)^2}}{n_A+n_B-2}

with degrees of freedom (df): \(df = n_A + n_B - 2\).

  1. Welch t-statistic. If the variances of the two groups being compared are different (heteroscedasticity), it’s possible to use the Welch t-test, which is an adaptation of the Student t-test. The Welch t-statistic is calculated as follow :

t = \frac{m_A - m_B}{\sqrt{ \frac{S_A^2}{n_A} + \frac{S_B^2}{n_B} }}

where, \(S_A\) and \(S_B\) are the standard deviation of the the two groups A and B, respectively.

Unlike the classic Student’s t-test, the Welch t-test formula involves the variance of each of the two groups (\(S_A^2\) and \(S_B^2\)) being compared. In other words, it does not use the pooled variance \(S\).

The degrees of freedom of Welch t-test is estimated as follow :

df = (\frac{S_A^2}{n_A}+ \frac{S_B^2}{n_B})^2 / (\frac{S_A^4}{n_A^2(n_A-1)} + \frac{S_B^4}{n_B^2(n_B-1)} )

A p-value can be computed for the corresponding absolute value of t-statistic (|t|).

If the p-value is inferior or equal to the significance level 0.05, we can reject the null hypothesis and accept the alternative hypothesis. In other words, we can conclude that the mean values of group A and B are significantly different.

Note that, the Welch t-test is considered as the safer one. Usually, the results of the classical student’s t-test and the Welch t-test are very similar unless both the group sizes and the standard deviations are very different.

Related article

T-test in R

Version: Français

One Sample T-Test Formula (Prev Lesson)
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