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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.

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