We study the central problem in data privacy: how to share data with an analyst while providing both privacy and utility guarantees to the user that owns the data. In this setting, we present an estimation-theoretic analysis of the privacy-utility trade-off (PUT). Here, an analyst is allowed to reconstruct (in a mean-squared error sense) certain functions of the data (utility), while other private functions should not be reconstructed with distortion below a certain threshold (privacy). We demonstrate how chi-square information captures the fundamental PUT in this case and provide bounds for the best PUT. We propose a convex program to compute privacy-assuring mappings when the functions to be disclosed and hidden are known a priori and the data distribution is known. We derive lower bounds on the minimum mean-squared error of estimating a target function from the disclosed data and evaluate the robustness of our approach when an empirical distribution is used to compute the privacy-assuring mappings instead of the true data distribution. We illustrate the proposed approach through two numerical experiments.