The invention determines the population size and population overlap in data containing records on the unique entities without unique identifiers for the unique entities and having at least one common type of information with a known distribution of finite expectation by decomposing probabilistic calculations. The computer determines population overlap of unique entities between the data sets by subtracting a probabilistic incremental number of unique entities needed for a larger total number of values of the information with the known distribution from the data sets. The invention can also maintain the security of private data by allowing a remote computer where the original data is stored to download diagnostic and aggregation procedures from another computer over a network. The remote computer performs the functions on the data and forwards the results to the estimate processor computer over the network. The estimate processor determines population size and overlap from aggregate results and forwards this information back to the remote computer over the network. The invention also determines the overlap of three or more data sets by concatenating all combinations of the data sets and determining estimates for all subsets of the combinations of the data sets. The operations involve the cancellation of equivalent terms that have opposite signs.