[s1artibartfast]

I think you're right. The 3% number must be ignoring repeat sampling bias. This is basically the same issue as P hacking or false positives and medical testing.

You have one confidence margin for a single single measurement and a different confidence margin if you make 1 million measurements.

Let's say you can measure marble diameters and your tool has a calibrated standard deviation of 1 mm.

If you pull one marble and measure it to be 10 mm larger than expected, you can calculate the chance you are wrong using only the standard deviation of your measurement tool.

However, if you pull 1 million marbles and measure one to be 10 mm larger than expected, you need to take into account the number of marbles you have measured.