October 2012 Archives

Misinterpreting Exponential Growth

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Imagine that the plot below represents the number of users over time for your favorite web service. Can you find the moment where the service starts to pick up and become successful? 


If you're like most people, you suspect something happened around x=40. Maybe a change in CEO? Maybe a positive review at TechCrunch? Given any service and a plot that looks like this, you'll probably be able to find something to explain a noticeable change in behavior - from a stagnating service to a stellar growthYou may have also thought that the change is from mediocre growth to exponential growth. The term "hockey stick growth" may have come to your mind, if you're familiar with it. 

However, there is nothing different at x=40, or any other point for that matter. The plot merely shows an exponential growth throughout the whole range (y=2^x). The numbers just happen to become large enough at around x=40 for our visual system to start to think there is a change in the underlying model. If you zoomed in at any point of the plot, you would see the same behavior, as in the plot below (showing the same data, but with x from 0 to 25). 


I've seen this mistake happen over and over in different contexts. Exponential curves are very tricky to be analysed by visual inspection and it is really hard not to think something magical happens at some point. But this point is arbitrary - a change in the representation (as zooming in) changes the point - and I'm sure one could find a nice story that fits nicely and "explains" what happened at that point, especially if this story favors whoever is telling it.

So the next time you see a curve like those above, know that whatever was done right (or wrong) was done from the start.

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