To preserve privacy, we divide the data space into boxes, and instead of original data points, only store the corresponding boxes. In accordance with the current practice, the desired level of privacy is established by having at least k different records in each box, for a given value k (the larger the value k, the higher the privacy level).
When we process the data, then the use of boxes instead of the original exact values leads to uncertainty. In this paper, we find the (asymptotically) optimal subdivision of data into boxes, a subdivision that provides, for a given statistical characteristic like variance, covariance, or correlation, the smallest uncertainty within the given level of privacy.
In areas where the empirical data density is small, boxes containing k points are large in size, which results in large uncertainty. To avoid this, we propose, when computing the corresponding characteristic, to only use data from boxes with a sufficiently large density. This deletion of data points increases the statistical uncertainty, but decreases the uncertainty caused by introducing the privacy-related boxes. We explain how to compute an optimal threshold for which the overall uncertainty is the smallest.