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The Gaussian mechanism (GM) represents a universally employed tool forachieving differential privacy (DP), and a large body of work has been devotedto its analysis. We argue that the three prevailing interpretations of the GM,namely $(\varepsilon, \delta)$-DP, f-DP and R\'enyi DP can be expressed byusing a single parameter $\psi$, which we term the sensitivity index. $\psi$uniquely characterises the GM and its properties by encapsulating its twofundamental quantities: the sensitivity of the query and the magnitude of thenoise perturbation. With strong links to the ROC curve and thehypothesis-testing interpretation of DP, $\psi$ offers the practitioner apowerful method for interpreting, comparing and communicating the privacyguarantees of Gaussian mechanisms.

A unified interpretation of the Gaussian mechanism for differential privacy through the sensitivity index