T-Test Essentials: Definition, Formula and Calculation

Formula

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.

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

where,

• \(m_A\) 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): \(df = n_A + n_B - 2\).

2. 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 :

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

Related article

T-test in R