The Gaussian mechanism (GM) represents a universally employed tool for achieving differential privacy (DP), and a large body of work has been devoted to its analysis. We argue that the three prevailing interpretations of the GM, namely $(\varepsilon, δ)$-DP, f-DP and Rényi DP can be expressed by using a single parameter $ψ$, which we term the sensitivity index. $ψ$ uniquely characterises the GM and its properties by encapsulating its two fundamental quantities: the sensitivity of the query and the magnitude of the noise perturbation. With strong links to the ROC curve and the hypothesis-testing interpretation of DP, $ψ$ offers the practitioner a powerful method for interpreting, comparing and communicating the privacy guarantees of Gaussian mechanisms.