# Estimated standard deviation of difference between means in case of known and unknown variance? [duplicate]

I am referring to this document page 198-9:

I have troubles understanding the case with unequal variance.

In general it holds:

or on Wikipedia:

Therefore if we assume unequal variance, I would think like this:

$$s_{\bar{X}_T-\bar{X}_C}=\sqrt{s^2_T/n_1+s^2_C/n_2}$$

Howerver, they give this for the standard error:

I have troubles with deriving these two different formulas.

• I would recommend watching this video. khanacademy.org/math/ap-statistics/random-variables-ap/… I share your sentiment that finding sources for combining random variables is difficult, hopefully this provides some assistance. Mar 8 '20 at 20:14
• Thanks for your post, however that does not answer my question or help my. Furthermore please note that I am not interested in the distribution (mean and variance) of the sum (or difference) of two random variables, but my question refers to the variance of the difference of the means. So not the variance of the distribution of two random variables. Mar 8 '20 at 20:42