# How to incrementally update a Welch's $t$-Test?

Say I have two sample distributions where for each distribution I know:

• Number of Values: $$n_1, n_2$$.
• The means: $$m_1, m_2$$.
• Standard Deviation: $$s_1, s_2$$.

With this information I am able to calculate a $$t$$-value, $$t_s$$, using Welch's $$t$$-test (we assume the two distribution do not share the same variance).

If we were to add a single number to each sample distribution and recalculate the t-value, what would be the most efficient way to do it? I know based off: How to add and subtract values from an average? and incremental computation of standard deviation that it is possible to incrementally update the mean and standard deviation. I know I could plug in the formulas from each page to find a solution, however I was looking for something in terms of $$t_s$$. I am new to stats and I am wondering if there is a name for all of this that I can look up.