The Art of Thinking Clearly: Better Thinking, Better Decisions

See also Sunk Cost Fallacy (ch. 5); Cognitive Dissonance (ch. 50)

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61
WHY SMALL THINGS LOOM LARGE
The Law of Small Numbers
You sit on the corporate board of a retail company with 1,000 stores. Half of the
stores are in cities, the other half in rural areas. At the behest of the CEO, a
consultant conducted a study on shoplifting and is now presenting his findings.
Projected on to the wall in front are the names of the 100 branches that have the
highest theft rates compared to sales. In bold letters above them is his eye-
opening conclusion: ‘The branches with the highest theft rate are primarily in rural
areas.’ After a moment of silence and disbelief, the CEO is first to speak: ‘Ladies
and gentlemen, the next steps are clear. From now on, we will install additional
safety systems in all rural branches. Let’s see those hillbillies steal from us then!
Do we all agree?’
Hmmm, not completely. You ask the consultant to call up the 100 branches with
the lowest theft rates. After some swift sorting, the list appears. Surprise, surprise:
the shops with the lowest amount of shoplifting in relation to sales are also in
rural areas! ‘The location isn’t the deciding factor,’ you begin, smiling somewhat
smugly as you gaze around the table at your colleagues. ‘What counts is the size
of the store. In the countryside, the branches tend to be small, meaning a single
incident has a much larger influence on the theft rate. Therefore, the rural
branches’ rates vary greatly – much more than the larger city branches. Ladies
and gentlemen, I introduce you to the 
law of small numbers
. It has just caught you
out.’
T h e 
law of small numbers
is not something we understand intuitively. Thus
people – especially journalists, managers and board members – continually fall
for it. Let’s examine an extreme example. Instead of the theft rate, consider the
average weight of employees in a branch. Instead of 1,000 stores, we’ll take just
two: a mega-branch and a mini-branch. The big store has 1,000 employees; the
small store just two. The average weight in the large branch corresponds roughly
to the average weight of the population, say 170 pounds. Regardless of who is
hired or fired, it will not change much. Unlike the small store: in these cases, the
store manager’s colleague, if rotund or reedy, will affect the average weight

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tremendously.
Let’s go back to the shoplifting problem. We now understand why the smaller a
branch is, the more its theft rate will vary – from extremely high to extremely low.
No matter how the consultant arranges his spreadsheet, if you list all the theft
rates in order of size, small stores will appear at the bottom, large stores will take
up the middle – and the top slots? Small stores again. So, the CEO’s conclusion
was useless, but at least he doesn’t need to go overboard on a security system
for the small stores.
Suppose you read the following story in the newspaper: ‘Start-ups employ
smarter people. A study commissioned by the National Institute of Unnecessary
Research has calculated the average IQ in American companies. The result:
Start-ups hire MENSA material.’ What is your first reaction? Hopefully a raised
eyebrow. This is a perfect example of the 
law of small numbers
. Start-ups tend to
employ fewer people; therefore the average IQs will fluctuate much more than
those of large corporations, giving small (and new) businesses the highest and
lowest scores. The National Institute’s study has zero significance. It simply
confirms the laws of chance.
So, watch out when you hear remarkable statistics about any small entities:
businesses, households, cities, data centres, anthills, parishes, schools etc. What
is being peddled as an astounding finding is, in fact, a humdrum consequence of
random distribution. In his latest book, Nobel Prize winner Daniel Kahneman
reveals that even experienced scientists succumb to the 
law of small numbers.
How reassuring.