I got four measurements which I want to process with an RMS as follows: $$ r = \sqrt{\frac{a*m_1^2+b*m_2^2+c*m_3^2+d*m_4^2}{a+b+c+d}} $$
Now I get a value for $r$ which is nice. But to do further research on the result, I need to know what impacts the measurements $m_x$ had on the result. I would expect something like:
$m_{1,impact} = 15\%$
$m_{2,impact} = 48\%$
$m_{3,impact} = 7\%$
$m_{4,impact} = 30\%$
I'm not quite sure if there is maybe some standard procedure behind this that I'm missing now. Could you help me please to achieve what I want?
This is not an arithmetic mean, so the effect of each measurement is not a constant. In fact, each effect depends on all measurements and weights: $$r = \sqrt{w_1 m_1^2+w_2 m_2^2+w_3 m_3^2+w_4m_4^2} \implies \frac{\partial r}{\partial m_i} = \frac{1}{2r}\cdot 2w_im_i = \frac{w_im_i}{r} .$$