Welch t-test

What is Welch t-test

The Welch t-test is an adaptation of Student’s t-test. It is used to compare the means of two groups of samples when the variances are different.

Welch t-test formula

Welch t-statistic is calculated as follow :

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

  • A and B represent the two groups to compare.
  • \(m_A\) and \(m_B\) represent the means of groups A and B, respectively.
  • \(n_A\) and \(n_B\) represent the sizes of group A and B, respectively.
  • \(S_A\) and \(S_B\) are the standar deviation of the the two groups A and B, rspectively.

Unlike the classic Student’s t-test, 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 common variance.

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

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


Once the t value is determined, you have to read in the t table the critical value of Student’s t distribution corresponding to the significance level alpha of your choice (5%).


If the absolute value of the t statistic (|t|) is greater than the critical value, then the difference is significant. Otherwise it isn’t. The level of significance or (p-value) corresponds to the risk indicated by the t table for the calculated |t| value.

Online Student’s t-test calculator


An online Student’s t-test calculator is available here without any installation.



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