In this paper, we show a formal equivalence between histogram equalization and distribution-preserving quantization. We use this equivalence to connect histogram equalization to quantization for preserving anonymity under the k-anonymity metric, while maintaining distributional properties for data analytics applications. Finally, we make connections to mismatched quantization. These relationships allow us to characterize the loss in mean-squared error (MSE) performance of privacy-preserving quantizers that must meet distribution-preservation constraints as compared to MSE-optimal quantizers in the high-rate regime. Thus, we obtain a formal characterization of the cost of anonymity.