# 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

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

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