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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?

elyptikus
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1 Answers1

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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: $$r = \sqrt{w_1 m_1^2+w_2 m_2^2+w_3 m_3^2+w_4m_4^2} \implies \frac{\partial r}{\partial m_i} = \frac{1}{2r}\cdot 2w_im_i = \frac{w_im_i}{r} .$$

user1337
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  • Thanks a lot for this super fast answer! Is there a name for this value? And I guess in your answer is $\sum{w_x}=1$, right? – elyptikus May 20 '21 at 12:56
  • @dofrey I would just go with the name “partial derivative”. The sum of the weights is indeed 1. One thing you could do for interpretation is divide one partial derivative by another to get the relative importance. – user1337 May 21 '21 at 06:39
  • Makes sense. Thanks a lot! – elyptikus May 21 '21 at 10:02