The variance of Binomial distributions

Introduction

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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