Computing the Mean of Around Values

I understand the computation of the uncertainty of independent Around values to be based on the following first order approximation $$\delta f(x,y)=\sqrt{\left(\frac{\text{\delta f}}{\partial x}\text{\delta x}\right)^2+\left(\frac{\text{\delta f}}{\partial y}\text{\delta y}\right)^2}.$$

Thus I compute the uncertainty for the mean of around numbers to be $$\delta\text{Mean}\left[\left\{\text{Around}\left[x_1,\text{\delta x}_1\right],\text{...}\text{Around}\left[x_{\text{n1}},\text{\delta x}_n\right]\right\}\right]=\frac{\sqrt{\text{\delta x}_1^2+\text{...}+\text{\delta x}_n^2}}{n}$$

and this seems to work when the uncertainties are different. For instance Mean[{Around[1,0.5],Around[1,0.25}] = Around[1,0.28]. However, when the uncertainties are equal, Mathematica returns an Around value with uncertainty unchange. For instance I would expect Mean[{Around[1,0.5],Around[1,0.5}] to return Around[1,0.35] but instead it returns Around[1,0.5]. And if the values are different, I do get the expected result, thus the Mean[{Around[2,0.5],Around[1,0.5]}]=Around[1.5,0.35].

So I'm assuming Mathematica is treating Mean[{Around[1,0.5],Around[1,0.5]}] as two correlated inputs, but why and how would you control this?

• Looks like a bug to me - there is AroundReplace for the case of correlated quantities, so this should work as it is. My guess is that something inside Mean is simplifying the expression for the mean to (2*Around[1,0.5])/2 before doing any special handling of the Around quantities, which is causing the issue. The easiest fix for now is probably to use your own function to compute the mean, e.g. Total[#]/Length[#] &[{Around[1, 0.5], Around[1, 0.5]}] Oct 13, 2020 at 12:04
• Please make a report to [email protected] Oct 13, 2020 at 12:08
• I agree. There is probably a simplification happening which changes the calculation. I'll submit the note to [email protected]. Thanks for confirming. Oct 13, 2020 at 21:31
• BTW, in more recent releases this issue was resolved. Feb 8, 2022 at 16:56