How Not To Be Wrong - Jordan Ellenberg ****

In the preface to Jordan Ellenberg's chunky maths book (441 pages before the notes in the version I read) we are introduced to a hypothetical student moaning about having to work through a series of definite integrals and complaining 'When am I going to use this?' What Ellenberg sets out do is to show how we use mathematics all the time - and how important it is to understand it if we are not to get the wrong idea about the world. We'll see how well he does.

It was very interesting to read this book quite soon after Richard Nisbett's Mindware. Both cover how to interact with life better thanks to the support of mathematics. Nisbett drives from the psychology side and improving decision making, while this book drives from the maths. Perhaps surprisingly, How Not to be Wrong is the easier read of the two. Ellenberg has a delightful light touch and is often genuinely funny (it's important to read the footnotes, which Ellenberg, like Terry Pratchett, uses for a lot of his jokes).

Along the way he shows us the uses and risks of straight lines in forecasting and understanding data, the power (and danger) of using methods of inference, how to use expected value, the realities of regression to the mean and the interplay between correlation and causality, and some fascinating observations on why traditional statistics can be very misleading when it comes to public opinion. Here it is often not applied to either/or situations, and it's quite possible, for instance, for the public to both support the idea of cutting taxes while simultaneously supporting raising expenditure. Although there are a few cases where we lose the plot and the connection to the real world, mostly this all driven by real world examples - from lotteries where an appropriate strategy can result in big wins to the apparent prediction that everyone in America would be obese before the end of the century.

While I don't think is this as practical a book as Nisbett's, it is full of fascination for anyone who likes a bit of applied mathematics, but can't be bothered with the formulae - there is very little that is scary in that line here. What's more, if you have any exposure to scientists, this book contains by far the best explanation of p-values, what they really mean and where they are meaningless that I've ever seen. 

So would the student from the preface feel after reading this book that there's no need to complain? Satisfyingly for a book that doesn't limit us to predictable mathematical answers, the response is both yes and no. Yes, because it becomes very clear that maths is hugely useful in understanding the world and responding to it. No, because the vast majority of maths you will have suffered at school and may have suffered at university, isn't required here. At least 90 per cent of the content depends on probability and statistics, topics that are rarely covered well enough in the curriculum, given how important they are in getting a grip on reality.

Although it felt a bit too long and used US sports rather too often as examples for my liking, this is a book for anyone with an interest in the way that mathematics can give us a better understanding of what's really happening in our complex world.

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Review by Brian Clegg