I have a large dataset (the population) and a large subset of it (the sample) containing the same, continuous variables. The sample represents more than 90% of the population but is not random -- we would like the sample to be 100%, the difference is due to systematic problems.
I would like to measure the extent to which the distribution of each variable in the sample has shifted compared with its distribution in the population. This would help inform the analysts using our sample about whether they need to correct for any distortions our process might have introduced (the analysts are not allowed to access the population).
I've looked at using a two-tailed t-test and/or a Kolmogorov-Smirnov test (which was suggested in an answer here) but in both cases the setup is that you have two samples from an unknown population with a theoretical distribution. In my case I have the actual population, so I feel like there might be a better method.
I should add that these datasets have 100s of millions of rows with 1000s of variables, so computationally intensive options are not feasible. So the real problem is to identify one or more "canary in the coalmine" statistics that are cheap to calculate and will flag up a small subset of problematic variables for further analysis.