Why There Is No Such Thing as an Average War: The Problem with Averages
Suppose you’re on a bus with forty-nine other people. At the next stop, the heaviest person in American gets on. Question: By how much has the average weight of the passengers increased? Four percent? Five? Something like that? Suppose the bus stops again, and on gets Bill Gates. This time we are not concerned about weight. Question: By how much has the average wealth risen? Four percent? Five? Far from it!
Let’s calculate the second example quickly. Suppose each of fifty randomly selected individuals has assets of $54,000. This is the statistical middle value, the median. Then Bill Gates is added to the mix, with his fortune of around $59 billion. The average wealth has just shot up to $1.15 billion, an increase of more than two million percent. A single outlier has radically altered the picture, rendering the term “average” completely meaningless.
“Don’t cross a river if it is (on average) four feet deep,” warns Nassim Taleb, from whom I (Rolf Dobelli) have the above examples. The river can be very shallow for long stretches—mere inches—but it might transform into a raging torrent that is twenty feet deep in the middle—in which case you could easily drown. Dealing in averages is a risky undertaking because they often mask the underlying distribution—the way the values stack up.
Another example: The average amount of UV rays you are exposed to on a June day is not harmful to your health. But if you were to spend the entire summer in a darkened office, then fly to Barbados and lie in the sun without sunscreen for a week solid, you would have a problem—even though, on average over the summer, you were not getting more UV light than someone who was regularly outside.
All this is quite straightforward and maybe you were aware of it already. For example, you drink one glass of red wine for dinner every evening. That’s not a health issue. Many doctors recommend it. But if you drink no alcohol the entire year and on December 31 you gulp 356 glasses, which is equivalent to sixty bottles, you will have a problem, although the average over the year is the same.
Here’s the update: In a complex world, distribution is becoming more and more irregular. In other words, we will observe the Bill Gates phenomenon in ever more domains. How many visits does an average website get? The answer is: There are no average websites. A handful of sites (such as the New York Times, Facebook, or Google) garner the majority of visits, and countless other pages draw comparatively few. In such cases, mathematicians speak of the so-called power law. Take cities. There is one city on this planet with a population of more than thirty million: Tokyo. There are eleven cities with a population of between twenty and thirty million. There are fifteen cities with a population of between ten and twenty million. There are forty-eight cities between five and ten million inhabitants. And thousands (!) between one and five million. That’s a power law. A few extremes dominate the distribution, and the concept of average is rendered worthless.
What is the average size of a company? What is the average population of a city? What is an average war (in terms of deaths or duration)? What is the average daily fluctuation in the Dow Jones? What is the average cost overrun of construction projects? How many copies does an average book sell? What is the average amount of damage a hurricane wreaks? What is a banker’s average bonus? What is the average success of a marketing campaign? How many downloads does an average iPhone app get? How much money does an average actor earn? Of course you can calculate the answers, but it would be a waste of time. These seemingly routine scenarios are subject to the power law.
To use just the final example: A handful of actors take home more than $10 million per year, while thousands and thousands live on the breadline. Would you advise your son or daughter to get into acting since the average wage is pretty decent? Hopefully not—wrong reason.
In conclusion: If someone uses the word “average,” think twice. Try to work out the underlying distribution. If a single anomaly has almost no influence on the set, the concept is still worthwhile. However, when extreme cases dominate (such as the Bill Gates phenomenon), we should discount the term “average.” We should all take stock from novelist William Gibson: “The future is already here—it’s just not very evenly distributed.”
* Source: The Art of Thinking Clearly by Rolf Dobelli