The variance of Binomial distributions


Recently I’ve been working on a problem that besets researchers in corpus linguistics who work with samples which are not drawn randomly from the population but rather are taken from a series of sub-samples. These sub-samples (in our case, texts) may be randomly drawn, but we cannot say the same for any two cases drawn from the same sub-sample. It stands to reason that two cases taken from the same sub-sample are more likely to share a characteristic under study than two cases drawn entirely at random. I introduce the paper elsewhere on my blog.

In this post I want to focus on an interesting and non-trivial result I needed to address along the way. This concerns the concept of variance as it applies to a Binomial distribution.

Most students are familiar with the concept of variance as it applies to a Gaussian (Normal) distribution. A Normal distribution is a continuous symmetric ‘bell-curve’ distribution defined by two variables, the mean and the standard deviation (the square root of the variance). The mean specifies the position of the centre of the distribution and the standard deviation specifies the width of the distribution.

Common statistical methods on Binomial variables, from χ² tests to line fitting, employ a further step. They approximate the Binomial distribution to the Normal distribution. They say, although we know this variable is Binomially distributed, let us assume the distribution is approximately Normal. The variance of the Binomial distribution becomes the variance of the equivalent Normal distribution.

In this methodological tradition, the variance of the Binomial distribution loses its meaning with respect to the Binomial distribution itself. It seems to be only valuable insofar as it allows us to parameterise the equivalent Normal distribution.

What I want to argue is that in fact, the concept of the variance of a Binomial distribution is important in its own right, and we need to understand it with respect to the Binomial distribution, not the Normal distribution. Sometimes it is not necessary to approximate the Binomial to the Normal, and if we can avoid this approximation our results are likely to be stronger as a result.

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Binomial confidence intervals and contingency tests

Abstract Paper (PDF)

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.


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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