# Compute Standard Deviation for the Mean of Means

I need some help with the statistical analysis of an experiment. It would be great if you could cite on what literature you base your answer if possible.

For simplicity let us assume I have 3 repetitions of the same experiment $$exp_1$$, $$exp_2$$ and $$exp_3$$, where $$x^i$$, $$x^j$$ and $$x^k$$ are the same specific measurements across the repeats:

• $$exp_1$$: $$\ x_1^i, \ x_1^j, \ x_1^k$$
• $$exp_2$$: $$\ x_2^i, \ x_2^j, \ x_2^k$$
• $$exp_3$$: $$\ x_3^i, \ x_3^j, \ x_3^k$$

I want to compute the standard deviation for the means across measurements and repeats. My naive approach right now is to compute the means: $$m_1 = \text{mean}(x_1^i, x_1^j, x_1^k), \ m_2.., m_3..$$. Next I compute the mean of means $$m_{all} = \text{mean}(m_1, m_2, m_3)$$ and the standard deviation of means $$s_{all} = \text{std}(m_1, m_2, m_3)$$ but I think this is not the right way to do it and I should compute $$s_{all}$$ based on the individual $$s_1, \ s_2, \ s_3$$?