PlusMinus /: a_ ± σa_ + c_?NumericQ := N[(a + c) ± σa];
PlusMinus /: a_ ± σa_ + b_ ± σb_ := N[(a + b) ± Norm@{σa, σb}];
PlusMinus /: c_?NumericQ (a_ ± σa_) := N[c a ± Abs[c σa]];
PlusMinus /: (a_ ± σa_) (b_ ± σb_) := N[a b ± (a b Norm@{σa/a, σb/b})^2];
PlusMinus /: (a_ ± σa_)^c_?NumericQ := N[a^c ± Abs[a^c σa/a]];