calculating mean and SE from two means and two SEs

A drug trial is conducted identically in two different groups that are essentially similar at baseline. Can I obtain a joint mean and a joint SE?

For example, Group A has placebo group of n=224 mean of 37.6 SE of 3.3 Group B has n=236 mean of 30.9 SE of 3.0 What is their combined mean and SE and how do you do it?

• Why do you want to combine Group A and Group B? Commented Apr 7, 2017 at 20:08
• Possible duplicate of Combining two covariance matrices which includes this as a special.case. Commented Apr 8, 2017 at 9:28

The mean will be the weighted average of the individual means. Call the combined group $X$ and let $N=N_A+N_B$. Essentially you need the sum of observations which can you write in terms of $\mu_A$ and $\mu_B$. $$\mu_X = \frac{1}{N}\sum_i x_i = \frac{1}{N_A+N_B}(N_A\mu_A+N_B\mu_B)$$ To compute the sample variance you do the exact same procedure, this time you need the sum of squared observations which you can write in terms of $s_A^2$ and $s_B^2$.
For $A$ (and similarly for $B$), $$\sum_i a_i^2 = (N_A-1)s_A^2+N_A\mu_A^2.$$ Then we are ready to compute: \begin{align*} s_X^2&=\frac{1}{N - 1}\left(\sum_i x_i^2 - N\mu_X^2\right)\\ &=\frac{1}{N - 1}\left((N_A - 1)s_A^2 + N_A\mu_A^2 + (N_B - 1)s_B^2 + N_B\mu_B^2 - N\mu_X^2\right) \end{align*}