Computing the Mean of Around Values

Question

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?