A Comprehensive Comparison of the Wald, Wilson, and adjusted Wilson Confidence Intervals for Proportions
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arXiv
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| Format: | Preprint |
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2025
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| author | Kahouadji, Nabil |
| author_facet | Kahouadji, Nabil |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_10223 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Comprehensive Comparison of the Wald, Wilson, and adjusted Wilson Confidence Intervals for Proportions Kahouadji, Nabil Methodology Computation 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. |
| title | A Comprehensive Comparison of the Wald, Wilson, and adjusted Wilson Confidence Intervals for Proportions |
| topic | Methodology Computation |
| url | https://arxiv.org/abs/2508.10223 |