We consider the problem of the private release of statistics (like aggregate payrolls) where it is critical to preserve the contribution made by a small number of outlying large entities. We propose a privacy formalism, per-record zero concentrated differential privacy (PzCDP), where the privacy loss associated with each record is a public function of that record's value. Unlike other formalisms which provide different privacy losses to different records, PzCDP's privacy loss depends explicitly on the confidential data. We define our formalism, derive its properties, and propose mechanisms which satisfy PzCDP that are uniquely suited to publishing skewed or heavy-tailed statistics, where a small number of records contribute substantially to query answers. This targeted relaxation helps overcome the difficulties of applying standard DP to these data products.