# Locally Private Mean Estimation: Z-test and Tight Confidence Intervals

arXiv: https://arxiv.org/abs/1810.08054 · HTML: https://mlsanity.com/paper/1810.08054/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2018.

## Built on by (most cited first, 15 of 15)

- [Covariance-Aware Private Mean Estimation Without Private Covariance Estimation](https://mlsanity.com/paper/2106.13329/) (2021)
- [DP-PCA: Statistically Optimal and Differentially Private PCA](https://mlsanity.com/paper/2205.13709/) (2022)
- [Local Differential Privacy Is Equivalent to Contraction of Eγ-Divergence](https://mlsanity.com/paper/2102.01258/) (2021)
- [Private Prediction Sets](https://mlsanity.com/paper/2102.06202/) (2021)
- [Contraction of Locally Differentially Private Mechanisms](https://mlsanity.com/paper/2210.13386/) (2022)
- [Non-parametric Differentially Private Confidence Intervals for the Median](https://mlsanity.com/paper/2106.10333/) (2021)
- [Nonparametric extensions of randomized response for private confidence sets](https://mlsanity.com/paper/2202.08728/) (2022)
- [Federated Calibration and Evaluation of Binary Classifiers](https://mlsanity.com/paper/2210.12526/) (2022)
- [Resampling methods for private statistical inference](https://mlsanity.com/paper/2402.07131/) (2024)
- [Collaborative Mean Estimation over Intermittently Connected Networks with Peer-To-Peer Privacy](https://mlsanity.com/paper/2303.00035/) (2023)
- [Privacy Preserving Semi-Decentralized Mean Estimation over Intermittently-Connected Networks](https://mlsanity.com/paper/2406.03766/) (2024)
- [Privacy-Preserving Conformal Prediction Under Local Differential Privacy](https://mlsanity.com/paper/2505.15721/) (2025)
- [How Much is Left? LLMs Linearly Encode Their Remaining Output Length](https://mlsanity.com/paper/2607.05316/) (2026)
- [DPpack: An R Package for Differentially Private Statistical Analysis and Machine Learning](https://mlsanity.com/paper/2309.10965/) (2023)
- [Confidence Calibration of Deep Learning Systems](https://mlsanity.com/paper/2608.12100/) (2026)

## Abstract (arXiv)

This work provides tight upper- and lower-bounds for the problem of mean estimation under ε-differential privacy in the local model, when the input is composed of n i.i.d. drawn samples from a normal distribution with variance σ. Our algorithms result in a (1-β)-confidence interval for the underlying distribution's mean μ of length tilde Oleft( fracσ √log(frac 1 β)ε√ n right). In addition, our algorithms leverage binary search using local differential privacy for quantile estimation, a result which may be of separate interest. Moreover, we prove a matching lower-bound (up to poly-log factors), showing that any one-shot (each individual is presented with a single query) local differentially private algorithm must return an interval of length Omegaleft( fracσ√log(1/β)ε√nright).