What is 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.’ 

The 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 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.

Expectations – HANDLE WITH CARE

Similar Biases:
Exponential Growth – STUMPED BY A SHEET OF PAPER

Effort Justification – HURTS SO GOOD

The above article is from the book The Art of Thinking Clearly by Rolf Dobelli. The article is only for educational and informative purposes to explain and understand cognitive biases. It is a great book, definitely worth a read!