### Introduction

*How can we calculate confidence intervals on a property like ***sentence length** (as measured by the number of words per sentence)?

You might want to do this to find out whether or not, say, spoken utterances consist of shorter or longer sentences than those found in writing.

The problem is that the average number of words per sentence is not a probability. If you think about it, this ratio will (obviously) equal or exceed 1. So methods for calculating intervals on probabilities won’t work without recalibration.

**Aside:** You are most likely to hit this type of problem if you want to plot a graph of some non-probabilistic property, or you wish to cite a property with an upper and lower bound for some reason. Sometimes expressing something as a probability does not seem natural. However, it is a good discipline to *think* in terms of probabilities, and to convert your hypotheses into hypotheses about probabilities as far as possible. As we shall see, this is exactly what you have to do to apply the Wilson score interval.

Note also that just because you want to calculate confidence intervals on a property, you also have to consider whether the property is freely varying when expressed as a probability.

The Wilson score interval (*w*⁻, *w*⁺), is a robust method for computing confidence intervals about probabilistic observations *p*.

Elsewhere we saw that the Wilson score interval obtained an accurate approximation to the ‘exact’ Binomial interval based on an observed probability *p*, obtained by search. It is also well-constrained, so that neither upper nor lower bound can exceed the probabilistic range [0, 1].

But the Wilson interval is based on a probability. In this post we discuss how this method can be used for other quantities.

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