I have a measurement systems that outputs $X_i + dX_i$ measurements. I'm trying to figure out the most correct way of estimating quality of measured device. The relative error $dX_i/X_i$ is normally quite large - ~40%.
The main figures for me are the median and 90th, 95th, 99th percentiles.
How do the measurement errors $dX_i$ propagate into the percentile error?
So you know two numbers - an estimate of the quantity itself $X_i$ and an estimate of the uncertainty $\Delta X_i$ (perhaps let's not use small $d$ to avoid confusion with differentials).
It is important for you to know the exact specifications of the measurement device. Many measurement devices are variance-limited, which means that the uncertainty is dominated by the stochastic noise of the instrument and is thus random. In this case, you should be able to find out what probability distribution the noise takes, thus recover the underlying probability distribution. For the simplest example, consider a Gaussian distributed noise, where $X_i$ is an estimate of the mean and $\Delta X_i$ is the estimate of the standard deviation. You can use this information to reconstruct the Gaussian distribution of the noise and then compute whatever percentile of that distribution you are interested in. Note that many devices do not follow Gaussian distribution, and you may be very wrong if you approximate a highly skewed distribution using Gaussian.
However, it is important to know that many instruments are bias-limited. Take for example a regular roulette meter used to measure the size of your work desk. No matter how many times you repeat the measurement, you will not get accuracy below ~0.5mm, because the instrument is not designed to measure that fine. In this case, your $\Delta X_i$ is not the standard deviation of the data, but (an upper bound of) the bias of the instrument. It is meaningless to attempt to convert this quantity to percentiles as it is not random, and is related to the instrument and not to the data.
Naturally, all measurements have some bias and some variance, so it is important to know the specs prior to addressing any statistical questions
This looks like a good summary of my second point.
