There is no reason to assume a normal distribution. If you have a tool that measures to a precision of 2 decimal places, you have no information about what the distribution of the third decimal place might be.
This is correct, which is why intervals don't choose an interpretation of the region of uncertainty.
If you do have reason to interpret the uncertainty as normally distributed, you can use that interpretation to narrow operations on two intervals based on your acceptable probability of being wrong.
But if the interval might represent, for example, an unknown but systematic bias, then this would be a mistake. You'd want to use other methods to determine that bias if you can, and correct for it.
> There is no reason to assume a normal distribution.
There absolutely is with sane assumptions about how any useful measurement tool works. Gaussian distributions are going to approximate the actual distribution for any tool that's actually useful, with very few exceptions.
When fabricating, we'll often aim for the high end of a spec so you have material remaining to make adjustments. Most of our measurements actually follow double-tail or exponential distributions.
I'm sorry but if I give you a measuring tape that goes to 2 decimal places and you measure a piece of wood at 7.23 cm, when you get a more precise tape you have no information at all about what the third decimal place will turn out to be. It could be anywhere between 7.225 and 7.235, there is no expectation that it should be nearer to the centre. All true lengths between those two points will return you the same 7.23 measurement and none are more likely than any other given what you know.
