Many statistical methods rely on an underlying mathematical model of probability which is based on a simple approximation, one that is simultaneously well-known and yet frequently poorly understood.

This approximation is the Normal approximation to the Binomial distribution, and it underpins a range of statistical tests and methods, including the calculation of accurate confidence intervals, performing goodness of fit and contingency tests, line-and model-fitting, and computational methods based upon these. What these methods have in common is the assumption that the likely distribution of error about an observation is Normally distributed.

The assumption allows us to construct simpler methods than would otherwise be possible. However this assumption is fundamentally flawed.

This paper is divided into two parts: fundamentals and evaluation. First, we examine the estimation of error using three approaches: the ‘Wald’ (Normal) interval, the Wilson score interval and the ‘exact’ Clopper-Pearson Binomial interval. Whereas the first two can be calculated directly from formulae, the Binomial interval must be approximated towards by computational search, and is computationally expensive. However this interval provides the most precise significance test, and therefore will form the baseline for our later evaluations.

We consider two further refinements: employing log-likelihood in computing intervals (also requiring search) and the effect of adding a correction for the transformation from a discrete distribution to a continuous one.

In the second part of the paper we consider a thorough evaluation of this range of approaches to three distinct test paradigms. These paradigms are the single interval or 2 × 1 goodness of fit test, and two variations on the common 2 × 2 contingency test. We evaluate the performance of each approach by a ‘practitioner strategy’. Since standard advice is to fall back to ‘exact’ Binomial tests in conditions when approximations are expected to fail, we simply count the number of instances where one test obtains a significant result when the equivalent exact test does not, across an exhaustive set of possible values.

We demonstrate that optimal methods are based on continuity-corrected versions of the Wilson interval or Yates’ test, and that commonly-held assumptions about weaknesses of χ² tests are misleading.

Log-likelihood, often proposed as an improvement on χ², performs disappointingly. At this level of precision we note that we may distinguish the two types of 2 × 2 test according to whether the independent variable partitions the data into independent populations, and we make practical recommendations for their use.

### Introduction

Estimating the error in an observation is the first, crucial step in inferential statistics. It allows us to make predictions about what would happen were we to repeat our experiment multiple times, and, because each observation represents a sample of the population, predict the true value in the population (Wallis 2013).

Consider an observation that a proportion *p* of a sample of size *n* is of a particular type.

For example

- the proportion
*p* of coin tosses in a set of *n* throws that are heads,
- the proportion of light bulbs
*p* in a production run of *n* bulbs that fail within a year,
- the proportion of patients
*p* who have a second heart attack within six months after a drug trial has started (*n* being the number of patients in the trial),
- the proportion
*p* of interrogative clauses *n* in a spoken corpus that are finite.

We have one observation of *p*, as the result of carrying out a single experiment. We now wish to infer about the future. We would like to know how reliable our observation of *p* is without further sampling. Obviously, we don’t want to repeat a drug trial on cardiac patients if the drug may be adversely affecting their survival.

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