# A Comprehensive Comparison of the Wald, Wilson, and adjusted Wilson Confidence Intervals for Proportions

arXiv: https://arxiv.org/abs/2508.10223 · HTML: https://mlsanity.com/paper/2508.10223/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Aug 2025.

## Built on by (most cited first, 2 of 2)

- [AdaPlanBench: Evaluating Adaptive Planning in Large Language Model Agents under World and User Constraints](https://mlsanity.com/paper/2606.05622/) (2026)
- [Estimating Uncertainty in Classifier Performance with Applications to Large Language Models and Nested Data](https://mlsanity.com/paper/2606.26422/) (2026)

## Abstract (arXiv)

The standard confidence interval for a population proportion covered in the overwhelming majority of introductory and intermediate statistics textbooks surprisingly remains the Wald confidence interval despite having a poor coverage probability, especially for small sample sizes or when the unknown population proportion is close to either 0 or 1. Using the mean coverage probability, and for some sample sizes, Agresti and Coull showed not only that the 95% Wilson confidence interval performs better, but also showed that 95% adjusted Wilson of type 4 confidence interval, obtained by simply adding four pseudo-observations, outperforms both the Wald and the Wilson confidence intervals. In this paper, we introduce a rainbow color code and pixel-color plots as ways to comprehensively compare the Wald, Wilson, and adjusted-Wilson of type ε confidence intervals across all sample sizes n=1, 2, dots, 1000, population proportion values p=0.01, 0.02, dots, 0.99, and for the three typical confidence levels. We show not only that adding 3 (resp., 4 and 6) pseudo-observations is the best for the 90% (resp., 95% and 99%) adjusted Wilson confidence interval, but it also performs better than both the 90% (resp., 95% and 99%) Wald and Wilson confidence intervals.