Estimating experimental resolution of an instrument from data Say I have measured some quantity $x$ with an instrument producing a set of observations $x_i$. I don't have the specifications of my instrument, and I have to estimate the experimental resolution of it, how could I do it? My first guess would to take just
$$
R = \min_{i \neq j}|x_i-x_j|
$$
is there a better guess?

Thought experiments can be useful. Many
real experiments might have benefitted from additional prior thought.
As a thought experiment, suppose you have two objects with true 'sizes' $\mu_1$ and $\mu_2$ (where $\mu_1 <\mu_2)$ respectively, with $\delta = \mu_2 - \mu_1.$ If it is important for you to detect a discrepancy of size $\delta,$ then you might say $\delta$ is the desired "resolution."
Now suppose you have 1000 measurements on Item 1 and 1000 measurements on Item 2. Measurements are centered on the true $\mu_i$'s with standard deviation $\sigma.$  To be specific, let's suppose
$\mu_1 = 100, \mu_2 = 103, \sigma =1.$ (Using R.)
set.seed(2022)
x1 = rnorm(1000, 100, 1)
x2 = rnorm(1000, 103, 1)

If the measurements are all put together into one
sample of 2000, then we get the histogram below.

R code for figure:
x = c(x1, x2)
hist(x, prob=T, col="skyblue2")
 curve(.5*dnorm(x, 100, 1)+.5*dnorm(x,103,1),
  add=T, col="brown", lwd=2)

The result is a 'mixture distribution'. Notice
that we can see a dip in the center of the density
curve, which indicates that a difference in means
of $3\sigma = 3$ is large enough for us to see
that there are two distinct items with different means.
A difference in means of $2\sigma = 2$
would not be quite large enough to give a bimodal
mixture distribution. (The center of the
density would show a "flat spot," but there would be no dip. See end note.)
So, if we have a difference as large as $3\sigma,$
the we can "resolve" the distribution of measurements to detect that we are looking at
two items of different sizes.
Roughly speaking, for measurements that are normally distributed, we might say that we need
the standard deviations of the measurements to be about
a third of the difference in sizes for good resolution of the difference.

Notes: (1) Here is a mixture of two normal distributions in which the means are exactly two standard deviations apart. For
clarity, larger samples and more histogram bins were used.

(2) If you want to read more about mixture distributions, perhaps this Wikipedia page is a good place to start.
