How does one tell if a dataset is missing data at random? I've been reading up on how to impute missing values, and was wondering what techniques can be used to tell if data is really missing at random or systematically.
You can't tell, absolutely, at least, not from statistics.
You can compare the cases that are missing to those that are not missing on any variables that are present in both, but there still could be other things that aren't in the data set. For a simple example, suppose your data set consists only of two variables "race/ethnicity" and "income". You could see if the proportion of different ethnicities that are missing are all similar, but people could be (and quite likely are) skipping the income question because of other things.
The only way to tell for sure that data are missing completely at random or missing at random is if you know why they are missing. In my experience, this sometimes lets you conclude they are MCAR - when it's some documented computer glitch, for instance and sometimes lets you tell they are NOT missing at random, but does not let you conclude they are MAR.