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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 America gets on. Question: by how much has the
average
weight of the passengers increased? Four per cent? 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 per cent? 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 per cent. 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 have the above examples. The river can be very shallow – mere
inches –
for long stretches, 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 outside
regularly.
All this is quite straightforward and maybe you were aware of it already. For
example, you drink one glass of red wine at dinner every evening. That’s not a
health issue. Many doctors recommend it. But if you drink no alcohol the entire
year and on Dec 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 30 million: Tokyo. There are 11 cities with a population of between 20 and
30 million. There are 15 cities with a population of between 10 and 20 million.
There are 48 cities with between 5 and 10 million inhabitants. And thousands (!)
between 1 and 5 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.’
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