Adam Gartenberg's Blog

Business Analytics and Optimization, IBM and Social Marketing

Never underestimate - or misinterpret - the exponential function


"Crime in the city doubles this decade"

The human brain is a funny thing.  You toss a ball to a four year old, and he will do differential calculus in his head, without even knowing it, to reposition his body and hands to more or less line up with where the ball will be.  But present someone with findings centered on the exponential function, and most people simply can't properly assimilate the implications..

If you read the above headline about crime doubling in your local newspaper, would you be worried?  Would the local police commissioner or even the mayor still be in their jobs?  It certainly would make a lot of alarm bells go off.  

But what about this headline:
"Crime in the city has grown 7% a year over past decade"

Would those same alarm bells have gone off?  For most people, the answer would be no.  And yet, a 7% compound annual growth rate is the same thing as doubling in 10 years.

Ian Randall pointed me to the video below in response to one of my other "Big Law of Small Numbers" posts, and I'm glad I gave it a look.  The speaker, Dr. Albert A. Bartlett, is a professor emeritus of Physics at Univ of Colorado-Boulder, and in the video explains in a very easy to understand manner and with compelling examples exactly why understanding the exponential function is so important - and how it is so easily misunderstood.



One of the examples he cites relates to growth for the city of Boulder.  City managers were asked what they thought an appropriate and acceptable growth rate for the city would be.  Several responded with growth rates at around the US average, which is ~1%.  Some were higher, with at least one city manager saying that he thought 5% would be a reasonable figure.  While it may sounds good on the surface -- as Prof. Bartlett says, who could argue with a good, healthy all-american growth rate like 5% -- looking at it a different way produces a vastly different picture.  A 5% growth rate means that over the course of a single lifetime - 70 years - the city would have a population 32 times higher than today.  Or, as he translates this to practical terms - if the city has one overtaxed sewage treatment plant today, they would need 32 overtaxed sewage treatment plants by the end of that time period just to keep up.

His basic argument, as laid out in the crime rate examples above, is that while the human mind doesn't seem to be very good at translating annual growth rate percentages back into real world implications, we do have a much better grasp at the idea of something doubling over a time period.  And the equation to make that conversion is relatively simple.  Just divide 70 by the growth rate, and that will tell you around how many years it will take to double.  So that 7% annual increase in crime equates to a doubling in crime in 10 years; a 5% annual growth in population means population will double every 14 years, or will be 32 times larger over the course of that same 70 years (2^5).

Of course, this has implications for those of us in marketing, as well.  I spend a lot of time preparing, presenting, and being presented with growth numbers.  How fast is my business growing compared to that of my competitors? Which industries are growing faster than others?  Will we do more business in India or Germany next year?  How about in 10 years?  

Are you properly interpreting the numbers you are being shown?  And when you're trying to convince someone to make an investment or to approve your business plan, are you putting the numbers in a light that will be best understood by the people to whom you are presenting?

One more personal note on this subject.  The professor in my core statistics course at business school began the semester by making us watch a video of the space shuttle Challenger disaster.  Back in the pre-9/11 days, this was the "where were you when..." moment for my generation, and there was a lot of uncomfortable silence as we watched the countdown - knowing full well what would happen - and on for several minutes after the breakup.  His reason for showing it to us was simple - one of the major contributing factors was that analysis on the impact of cold temperatures on the O-rings was modeled on a linear scale rather than a logarithmic scale.  Had they properly extrapolated the record cold temperatures that morning against the expected failure point using a logarithmic model, they would have come to a very different conclusion about the likelihood of the O-ring failing upon takeoff.  Even when not dealing with potential life-and-death analysis, it's a lesson I haven't forgotten.