Plotting the Wilson Distribution

While the Wilson score interval has served as the foundation for statistical inference throughout this book, its underlying probability density function distribution has remained unexplored. This chapter fills that gap by explaining a method to visualise and analyse the probability density function distribution of Wilson interval bounds, revealing fundamental properties that challenge common statistical assumptions. The chapter explains how to plot Wilson distributions by applying computed differentiation techniques to the Wilson function for bounds, noting that the Wilson score interval formula is actually a cumulative density function. By varying the error level α rather than p or n , the method exposes how the Wilson interval behaves across all possible confidence levels simultaneously. Key findings reveal that Wilson distributions deviate dramatically from Normal expectations, particularly near probability boundaries. As observed proportions approach 0 or 1, distributions become increasingly skewed, with modes shifting away from observed values. For very small sample sizes, the interaction between wide intervals and probability boundaries creates surprising bimodal ‘wisdom tooth’ distributions that fundamentally challenge bell curve assumptions. The chapter demonstrates that these apparently anomalous behaviours result from the bounded nature of probability space itself, not from flaws in the Wilson method. When transformed to the logit scale, Wilson distributions closely approximate Normal distributions, providing theoretical justification for logit-Wilson regression approaches and explaining why logistic transformations prove effective for bounded data.