The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty

CHAPTER 12

Who Was Jensen and Why Wasn’t He Equal?

THE NUTS AND BOLTS OF THE STRONG FORM OF THE FLAW OF AVERAGES

 

 

 

The Strong Form of the Flaw of Averages, which is known to mathematicians as Jensen’s Inequality, masks both risks and opportunities. Familiarity with the concept will help you avoid the former and seize the latter. In the last chapter we saw that the Flaw of Averages cuts both ways. When you plug averages of uncertain values into your plans, they sometimes overestimate the average outcome and sometimes underestimate the average outcome. What follows are some general rules that characterize these two situations.

Spreadsheets with Uncertain Inputs

The prevalence of the Flaw of Averages is due to the fact that millions of spreadsheet users plug their best guesses of uncertain numbers into their models, naively believing that they are getting the best outputs. However, the Strong Form of the Flaw of Averages isn’t news to mathematicians. They have known it for over a century as JENSEN’S INEQUALITY. Of course, with a name like that, no wonder no one else has heard of it.

A mathematician, by the way, would refer to a typical spreadsheet model with inputs, outputs, and formulas connecting them as a FUNCTION. And if some of the inputs were uncertain numbers, a mathematician would call it a FUNCTION OF RANDOM VARIABLES.

Therefore, the millions of spreadsheet models in the world with uncertain inputs are actually FUNCTIONS OF RANDOM VARIABLES. So if you use spreadsheets, ...