In this paper, we present a Distribution-Preserving Voice Anonymization technique, as our submission to the VoicePrivacy Challenge 2020. We observe that the challenge baseline system generates fake X-vectors which are very similar to each other, significantly more so than those extracted from organic speakers. This difference arises from averaging many X-vectors from a pool of speakers in the anonymization process, causing a loss of information. We propose a new method to generate fake X-vectors which overcomes these limitations by preserving the distributional properties of X-vectors and their intra-similarity. We use population data to learn the properties of the X-vector space, before fitting a generative model which we use to sample fake X-vectors. We show how this approach generates X-vectors that more closely follow the expected intra-similarity distribution of organic speaker X-vectors. Our method can be easily integrated with others as the anonymization component of the system and removes the need to distribute a pool of speakers to use during the anonymization. Our approach leads to an increase in EER of up to $19.4\%$ in males and $11.1\%$ in females in scenarios where enrollment and trial utterances are anonymized versus the baseline solution, demonstrating the diversity of our generated voices.