I have the list of experimental values: $$\{w_i \pm \Delta w_i\},$$ where $w_j$ is a mean value and $\Delta w_i$ is an error.

I want to calculate the second list $\{a_i \pm \Delta a_i\}$ according to the rule $$a_i = \frac{w_i / A_i}{\sum_j w_j / A_j},$$ where $A_i$ is some constant for $i$.

How to define errors $\Delta a_i$? Is it the right way?

$$\Delta a_i = \frac{1}{A_i} \frac{\Delta w_i |\sum_j w_j/A_j| + |w_i|\sum_j \Delta w_j/A_j}{(\sum_j w_j/A_j)^2}$$

  • $\begingroup$ I don't have the energy to check your workings but the relevant maths is here. Alternatively you can find a list of free software here that computes such things for you. $\endgroup$
    – lemon
    Commented Sep 15, 2014 at 8:36
  • $\begingroup$ It depends on what kind of uncertainties you are dealing with. Due you assume that it is normal distributed, where the error is (a multiple of) the standard deviation, or do you assume 100% confidence intervals? $\endgroup$
    – fibonatic
    Commented Dec 23, 2014 at 5:13
  • $\begingroup$ In your 1st equation , I think you have mistaken wi for wj. $\endgroup$ Commented May 3, 2016 at 12:37

1 Answer 1


According to link from foregoing comment $a_i$ measurments are inderect, so errors are

$$\Delta a_{i}=\sqrt{\sum_{k}\left(\frac{\partial a_{i}}{\partial w_{k}}\Delta w_{k}\right)^{2}}.$$

After calculations it yields $$\Delta a_{i}= \frac{1}{\left(\sum_{j}\frac{w_{j}}{A_{j}}\right)^{2}A_{i}}\sqrt{\sum_{k}\left( \left[\delta_{ik}\sum_{j}\frac{w_{j}}{A_{j}}-\frac{w_{i}}{A_{k}} \right] \Delta w_{k} \right)^{2}}$$

  • $\begingroup$ What does inderect mean? Perhaps a typo for indirect, but surely the word you want there is independent? $\endgroup$
    – innisfree
    Commented Oct 12, 2015 at 2:24

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