Say I have some unknown constant X, and I repeat an experiment many times where I measure X, so by the end I have a set of measurements $X_i$ with some mean and variance. My question is: is there some way to tell from the data wether all the measurements were done independently, or is there some dependency/correlation between different measurements in the set?
Thanks in advance
You need to somehow model the factors that could induce dependence among the measurements.
For example, suppose that dependence can be induced by 1) time-varying environmental conditions; 2) different instruments used to perform the measurement; 3) different people performing the measurement. You could perform a linear regression of your measurements on 1) time, 2) dummy variables that identify the instruments being used, 3) dummy variables that identify the person performing the measurement. Then, you could test whether the regression coefficients on 1), 2) and 3) are significantly different from zero. If you find that any coefficient is different from zero, then you have found a factor that induces dependence among your measurements (i.e., the measurement are independent conditional on that factor, but not unconditionally).
If you think that time is the only relevant dimension that could influence your measurements, then you could check whether the measurements are serially correlated, and in case they are you need to take into account the fact that serial correlation may inflate the variance of the sample average of the measurement (see, e.g., the proof of a WLLN for correlated sequences).