Let's say I have a magnitude $\lambda=f(x,y)$ where $x$ and $y$ are quantities that I've experimentally measured so I have two series of values $ \{x_1, x_2, ... x_n\}$ and $\{y_1, y_2, ... y_n\}$. Each measure has it's own error $\delta_{x_i}$ or $\delta_{y_i}$. To calculate the error of $\lambda$ I will use $\delta_{\lambda_i}=\sqrt{\left(\frac{\partial f}{\partial x_i}\delta_{x_i}\right)^2+\left(\frac{\partial f}{\partial {y_i}} \delta_{y_i}\right)^2}$. This way I can obtain a value $\lambda_i$ for each $x_i$ $y_i$. To get the average I will use a weighted average such as $\overline{\lambda}=\frac{\sqrt{\left(\sum_{i=1}^n \lambda_i \cdot \omega_i\right)^2}}{\sum_{i=1}^n \omega_i}$ in which $\omega_i=\frac{1}{{\delta_{\lambda_i}}^2}$. And I've been told to use this formula to calculate the error of that average: $\delta_{\overline{\lambda}}=\frac{1}{\sqrt{\sum_{i=1}^n \omega_i}}$. Now what I want is to calculate the error contribution of $x$ and $y$ to $\overline{\lambda}$. To calculate the error contributions for each $\lambda_i$ I just use the following formula: $\delta_{\lambda_i \leftarrow x_i}=\frac{\left(\frac{\partial f}{\partial {x_i}} \delta_{x_i}\right)^2}{{\delta_\lambda}^2}$. If I do this for $x$ and $y$ and sum both for each measurement $i$, I get 1, which makes sense. But how can I do the same for the average? If I average this partial errors just as I did before they do not add up to 100% of the error, what would be the approach here?
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