I think you have a misunderstanding about sigma. When describing the measurement of a particular physical constant, the "standard deviation" is something that c

I think you have a misunderstanding about sigma. When describing the measurement of a particular physical constant, the "standard deviation" is something that changes as we get better at making measurements. It basically means "If all our assumptions (e.g. assumptions about how good our equipment is, uncertainties about other physical constants) are correct, then it is unlikely that we would have made the measurements we did if the true value is not within 2 standard deviations of the result we got". When a more accurate measurement is made, then "1 standard deviation" gets smaller, so we know the value better, but it's always true to say "we know the value to within a few standard deviations (given some assumptions made by experimenters)" . If it turns out the measurement was wrong by several standard deviations, then it's very likely that some assumptions were wrong.