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The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty Paperback – March 15, 2012
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As the recent collapse on Wall Street shows, we are often ill-equipped to deal with uncertainty and risk. Yet every day we base our personal and business plans on uncertainties, whether they be next month’s sales, next year’s costs, or tomorrow’s stock price. In The Flaw of Averages, Sam Savageknown for his creative exposition of difficult subjects describes common avoidable mistakes in assessing risk in the face of uncertainty. Along the way, he shows why plans based on average assumptions are wrong, on average, in areas as diverse as healthcare, accounting, the War on Terror, and climate change. In his chapter on Sex and the Central Limit Theorem, he bravely grasps the literary third rail of gender differences.
Instead of statistical jargon, Savage presents complex concepts in plain English. In addition, a tightly integrated web site contains numerous animations and simulations to further connect the seat of the reader’s intellect to the seat of their pants.
The Flaw of Averages typically results when someone plugs a single number into a spreadsheet to represent an uncertain future quantity. Savage finishes the book with a discussion of the emerging field of Probability Management, which cures this problem though a new technology that can pack thousands of numbers into a single spreadsheet cell.
Praise for The Flaw of Averages
“Statistical uncertainties are pervasive in decisions we make every day in business, government, and our personal lives. Sam Savage’s lively and engaging book gives any interested reader the insight and the tools to deal effectively with those uncertainties. I highly recommend The Flaw of Averages.”
―William J. Perry, Former U.S. Secretary of Defense
“Enterprise analysis under uncertainty has long been an academic ideal. . . . In this profound and entertaining book, Professor Savage shows how to make all this practical, practicable, and comprehensible.”
―Harry Markowitz, Nobel Laureate in Economics
- Print length416 pages
- LanguageEnglish
- PublisherWiley
- Publication dateMarch 15, 2012
- Dimensions6 x 1.3 x 8.9 inches
- ISBN-101118073754
- ISBN-13978-1118073759
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Editorial Reviews
From the Inside Flap
Praise for
THE FLAW OF AVERAGES
"Statistical uncertainties are pervasive in decisions we make every day in business, government, and our personal lives. Sam Savage's lively and engaging book gives any interested reader the insight and the tools to deal effectively with those uncertainties. I highly recommend The Flaw of Averages."
William J. Perry, former U.S. Secretary of Defense
"Enterprise analysis under uncertainty has long been an academic ideal. . . . In this profound and entertaining book, Professor Savage shows how to make all this practical, practicable, and comprehensible . . . the Distribution String . . . represents a major breakthrough in the communication of risk and uncertainty."
Harry Markowitz, Nobel Laureate in Economics
"This is a book written for laymen with enough interesting insights to engage even the most scholarly professional."
Douglas Hubbard, author of How to Measure Anything
"Sam Savage is the Edward Tufte of risk."
Matthew Raphaelson, Executive Vice President, Wells Fargo
A GROUNDBREAKING MUST-READ FOR ANYONE WHO MAKES BUSINESS DECISIONS IN THE FACE OF UNCERTAINTY
In The Flaw of Averages, Sam Savageknown for his creative exposition of difficult subjectsdescribes common avoidable mistakes in assessing risk in the face of uncertainty. He explains why plans based on average assumptions are wrong, on average, in areas as diverse as finance, healthcare, accounting, the war on terror, and climate change. Savage refers to anachronistic statistical jargon as Red Words, which he defines as things that may not be uttered in a singles bar. Instead, he presents complex concepts in plain English (Green Words), backed up by interactive simulations at www.FlawofAverages.com, which connect the seat of the intellect to the seat of the pants.
Savage also presents the emerging field of Probability Management aimed at curing the Flaw of Averages through more transparent communication of uncertainty and risk. Savage argues that this is a problem that must be solved if we are to improve the stability of our economy, and that we cannot repeat the recent mistakes of applying "steam era" statistics to "information age" risks.
From the Back Cover
Praise for
THE FLAW OF AVERAGES
"Statistical uncertainties are pervasive in decisions we make every day in business, government, and our personal lives. Sam Savage's lively and engaging book gives any interested reader the insight and the tools to deal effectively with those uncertainties. I highly recommend The Flaw of Averages."
William J. Perry, former U.S. Secretary of Defense
"Enterprise analysis under uncertainty has long been an academic ideal. . . . In this profound and entertaining book, Professor Savage shows how to make all this practical, practicable, and comprehensible . . . the Distribution String . . . represents a major breakthrough in the communication of risk and uncertainty."
Harry Markowitz, Nobel Laureate in Economics
"This is a book written for laymen with enough interesting insights to engage even the most scholarly professional."
Douglas Hubbard, author of How to Measure Anything
"Sam Savage is the Edward Tufte of risk."
Matthew Raphaelson, Executive Vice President, Wells Fargo
A GROUNDBREAKING MUST-READ FOR ANYONE WHO MAKES BUSINESS DECISIONS IN THE FACE OF UNCERTAINTY
In The Flaw of Averages, Sam Savageknown for his creative exposition of difficult subjectsdescribes common avoidable mistakes in assessing risk in the face of uncertainty. He explains why plans based on average assumptions are wrong, on average, in areas as diverse as finance, healthcare, accounting, the war on terror, and climate change. Savage refers to anachronistic statistical jargon as Red Words, which he defines as things that may not be uttered in a singles bar. Instead, he presents complex concepts in plain English (Green Words), backed up by interactive simulations at www.FlawofAverages.com, which connect the seat of the intellect to the seat of the pants.
Savage also presents the emerging field of Probability Management aimed at curing the Flaw of Averages through more transparent communication of uncertainty and risk. Savage argues that this is a problem that must be solved if we are to improve the stability of our economy, and that we cannot repeat the recent mistakes of applying "steam era" statistics to "information age" risks.
About the Author
SAM L. SAVAGE is a Consulting Professor of Management Science and Engineering at Stanford University, and a Fellow of the Judge Business School at the University of Cambridge.
Product details
- Publisher : Wiley; 1st edition (March 15, 2012)
- Language : English
- Paperback : 416 pages
- ISBN-10 : 1118073754
- ISBN-13 : 978-1118073759
- Item Weight : 1 pounds
- Dimensions : 6 x 1.3 x 8.9 inches
- Best Sellers Rank: #209,000 in Books (See Top 100 in Books)
- #32 in Microeconomics (Books)
- #1,530 in Investing (Books)
- Customer Reviews:
About the author

Dr. Sam L. Savage is author of The Flaw of Averages: Why We Underestimate Risk in the Face of Uncertainty (John Wiley & Sons, 2009, 2012), and Executive Director of ProbabilityManagement.org, a 501(c)(3) nonprofit devoted to the communication and calculation of uncertainty. The organization has received funding from Chevron, Kaiser Permanente, Highmark Health, Lockheed Martin, PG&E, and others. Harry Markowitz, Nobel Laureate in Economics, was a co-founding board member.
Dr. Savage received his Ph.D. in computational complexity from Yale University, then began his career as a mathematician at the General Motors Research Laboratory. From there he taught at the University of Chicago Graduate School of Business for 15 years before moving to Stanford in 1990. He is currently an Adjunct Professor in Civil and Environmental Engineering and is also a Fellow of the University of Cambridge Judge Business School.
Throughout his career, Dr. Savage has consulted widely to industry and served as an expert in litigation, and his writing reflects his experiences on both the academic and practical side of what he calls “the algebraic curtain separating managers from management science.”
Dr. Savage is the inventor of the Stochastic Information Packet (SIP), an auditable data array for conveying uncertainty, which served as a basis for the nonprofit's open SIPmath™ Standard. This standard has been adopted by software firms such as Frontline Systems and Lumina Decision Systems.
In his own words, Dr. Savage says: “I am not a dispassionate observer, but a promoter who models himself after that great medieval huckster, Fibonacci, who foisted Hindu-Arabic numerals on an unsuspecting Western Civilization in 1199. At ProbabilityManagement.org, we are trying to create the Hindu-Arabic Numerals of Uncertainty."
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DISTRIBUTIONS. At its core, the Flaw of Averages is an oversimplified view of reality by fixating on a single number. “If organizations are to conquer the Flaw of Averages, they must start dealing in probability distributions, not in single numbers.” A simple way to view a distribution is a histogram. Figure 9.8 shows the profit of “28 live-action films of a particular genre, which had been selected so as not to over-represent any single theme or star.” The average profit is $20 million, but the histogram reveals a much more insightful picture of the risk: 25% of the films lost more than $5 million.
DIVERSIFICATION. If you invested in making one film, you’d have a 25% chance of losing your shirt, but the author ran some simulations to determine that investing in a portfolio of only four films would reduce the risk of loss to 1%, with the average profit remaining $20 million.
MODELS. “The surest way I know to detect the Flaw of Averages is the build a small spreadsheet model of your business situation and simulate the uncertainties you face. A good model improves your intuition by connecting the seat of your intellect to the seat of your pants… The best models are those you no longer need because they have changed the way you think… Most large financial organizations spend millions of dollars on complex risk models… that may actually mask what’s going on… What they need are simple models that everyone understands.”
UNCERTAINTY VERSUS RISK. “The terms ‘uncertainty’ and ‘risk’ are often used interchangeably, but they shouldn’t be… I consider uncertainty to be an objective feature of the universe, whereas risk is in the eye of the beholder.” Some people are more risk-averse than others when faced with the same uncertainty.
INTERRELATED UNCERTAINTIES. “People also use the term Statistical Dependence when discussing interrelated uncertainties, but this phrase implies that one uncertainty depends on another, when in fact the relationship may be mutual. So I will stick to interrelated uncertainties instead… Scatter plots are my favorite way to grasp the interrelationships between uncertain numbers.”
INFORMATION. “Information is the complement of uncertainty; that is, for every uncertainty, there is information that would reduce or remove it. Sometimes such information is cheap, and sometimes it is unavailable at any price… Information has no value at all unless it has the potential to change a decision.”
EXTREMES. “Did I mention where one would find the smallest average earlobes? Small towns, of course. The sizes of towns and earlobes have nothing to do with each other; it’s just that averages with small samples have more variability than averages over large samples.”
OPTIONS. “Recall the example of the gas well… that was worth $500,000 given the average price of gas but was worth $1 million on average. The increased value was due to the option not to pump in the event that the gas price was below the production cost. An opportunity such as this is known as a real option, and it is analogous to a call option on the gas, with a strike price equal to the pumping cost.”
FUTURES. “Given how terrible individuals are at estimating probabilities, it is a profound result that marketplaces usually get them right… For example, a classic 1984 paper by UCLA finance professor Richard Roll shows that the futures market in frozen concentrated orange juice is a better predictor of Florida weather than the National Weather Service… Although the futures markets predict the average future values of assets, as we have seen, option prices indicate the degree of uncertainty.”
ORGANIZATIONAL BOUNDARIES. David Cawlfield writes, “Of course the Flaw of Averages occurs everywhere, but often the flaws are small and go unnoticed, or the consequences are insignificant. At organizational boundaries the Flaw can cause real damage. This is where most of the cross-communication occurs between upper level managers, and involves decisions with large consequences. Lower in the organization, experts with the most knowledge about variability leave out the gory details about day-to-day ups and downs by communicating only the ‘big picture’ to the boss… An understanding of variability constitutes a large portion of the value of long work experience. The secret of successful Probability Management is learning to capture this understanding in a stochastic library.”
THE DECISION FOREST. “A firm introducing a new product lineup should use a portfolio approach to avoid risks due to both cannibalization and potential competitive products. A portfolio of pharmaceutical R&D projects would be designed with both competitive drugs and potential governmental regulation changes in mind. Most people [don’t take] true portfolio effects into account… Think of each project as a decision tree reflecting uncertainties that pertain only to that project… This results in a bunch of decision trees or a decision forest… But modeling all of the individual projects as decision trees is not enough, because, as discussed earlier, portfolio decisions must reflect he interrelationships between constituent parts. For this, I model global uncertainties such as the price of oil or political upheaval as the winds of fortune, which blow through the entire forest, influencing all the trees at once.”
MONTE CARLO. “A computational technique similar to the shaking of a ladder can test the stability of uncertain business plans, engineering designs, or military campaigns. Monte Carlo simulation, as it is known, bombards a model of the business, bridge, or battle with thousands of random inputs, while keeping track of the outputs. This allows you to estimate the chance that the business will go bust, the bridge will fall down, or the battle will be lost. The shaking forces you apply to the ladder are known as the input probability distribution and correspond to the uncertainty demand levels for your product, the magnitudes of potential earthquakes, or the sizes of the enemy forces you will encounter. The subsequent movements of the ladder are known as the output probability distribution and correspond to the profit of the business, the deflection of the bridge, or the number of casualties you suffer.”
DISTRIBUTION STRING. “The sum of the simulations of the parts is not the simulation of the whole… The key is a new computer data type called the Distribution String (DIST), which encapsulated distributions in a manner that allows them to be added together like numbers… In terms of the spinner, think of the DIST as consisting of the outcomes of one thousand spins, stuffed, like a genie in a bottle, into a single cell in your spreadsheet.”
SCENARIO LIBRARIES. In this example, risk is estimated across two banking divisions. “First, the chief probability officer (CPO) generates the distribution of housing market conditions. This is provided as a stochastic information packet (SIP)… One input SIP went in, and two output SIPs came out… They now form a scenario library unit with relationships preserved (SLURP), which maintains the dependence of each division on the housing market. The final step is just to add the output SIPs of the two divisions together to create the SIP of total profit… With SLURPs, the simulation of the sum does equal the sum of the simulations because it preserves the interrelationships… If you ignore the interrelationships, you would calculate the chance that both divisions lose money at the same time as 2/6 x 1/6 = 2/36 = 1/18.” With the consolidated method, “the chance of losing money is not 1 in 18 after all, but 1 in 3, six times greater!”
“If a CPO has no idea of a particular distribution, they should pull several different ones from the seat of their pants and see how the results differ under each. Today’s interactive simulations are so fast that you can actually enter probabilities as variables and discover at what point you would make different decisions.”
“But let’s not forget that risk is not inherently bad. In fact, risk, clearly presented and understood, is required for people to make investments. And if people don’t make investments, society pretty much grinds to a halt. Let’s also not forget that risk is in the eye of the beholder and that the risk attitude that a publicly traded firm should take on is the one anticipated by its shareholders. Unlike its employees, who may be more concerned about keeping their jobs than making profit, the shareholders, who are generally diversified across many other investments, want the firm to take the business risks that induced them to invest in the first place.”
Most likely you have never heard of Jensen's Inequality, and most likely you don't care. However, you should care, and The Flaw of Averages introduces why this concept poses profound implications to the way we tend to think about making decisions. The problem is that in thinking about the issues or opportunities we face and the decisions we exercise to address them, we often go through a kind of accounting process in which we consider the best case, most likely case, and worst case scenarios (or any number of scenarios) and the corresponding conditions that have to exist for each scenario to be realized. We let ourselves believe that those assumed conditions are averages that we can use as proxies for the full range of uncertainty we face. Our final conclusion is that the outcome of our analysis closely corresponds to the average real-world outcome and the extent of possible variation (if we get that far). Understanding that, we commit to action, often disastrously so.
Savage reveals the flaw in the traditional way of thinking by explaining the implications of Jensen's Inequality. In short, Jensen's Inequality says that in situations where the output we care about varies in a non-linear way to inputs (which is much of life), the outcome as a function of average inputs (the flawed traditional analytic approach) is NOT equal to the average outcome as a function of the inputs treated as they naturally vary. [For those mathematically inclined, if E() is an operator that determines the average of a sample, and f(Xi) is a function of inputs, then E( f(Xi ) ) 'does not equal f( E(Xi) ). For those not so mathematically inclined, don't worry. The Flaw of Averages is not a math book; rather, it is a book about making decisions and how math can be used constructively to support that process.]
The way around this failure is to use Monte Carlo simulation to consider simultaneously the effects of the range of the assumptions as they naturally vary on the outcome we care about. Instead of thinking about the outcome as a single point or a constellation of points representing exhaustive guesses about the future, we see the full range of potential outcomes and their likelihood as a distribution. We see the implications of our decisions and corresponding uncertainties as a picture and not a point. As a result, not only do we avoid never ending analysis paralysis, we gain a deeper appreciation for the effect of typically unconsidered outcomes, both good and bad, and are able to plan accordingly with contingencies and options.
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