## Wednesday, November 11, 2009

### The problems of percentages

This morning I read a Treehugger entry on how much greater New York City is now that their bicycling metric has increased by 26% over the last year. I don't have any problem with the article message, or it's tone, except that the writer states:
"[the increase] is a lot by any measure, though it is lower than the 35% increase in 2008."
That's an example of a poor understanding of mathematics. Why? Well, the increase in cycling went from 255 in 2008 to 321 in 2009, an increase of 66 points. The increase from 2007 to 2008 was 62 points. In other words, the two years' increases were roughly the same in terms of absolute value, but not in terms of percentage.

A percent growth means that the total value must increase in ever-greater amounts. In other words, a percent growth rate of 1% means that the initial value will double after 70 years; 2% will double in 35 years; 3% will double in 24 years; and so on. If the percent growth of 26% is continued, it would take only 3 years to double the initial value; in this case to go from 321 to 642 (which would be awesome).

On the other hand, a "mere" growth of 64 points per year (the average increase over the years) means an ever-decreasing annual percentage growth. Declining from the "lower" value of 26% it will sink to 17%, 14%, 12%, 11%, and so on in succeeding years (slowly approaching a 0% increase limit). Of course, the value will still have doubled by 2014 if the increment stays constant; two years slower than if the 26% growth rate was maintained.

Thinking in terms of percent growth is problematic, since thinking of percents in this way is actually implicitly implies that one is looking at growth rates. However, a growth of 0% is not the same as a growth rate of 0%. A growth of 0% from 2008 to 2009 would have meant a value of 255 in both years. However, a growth rate of 0% would mean that 2009 would have been 317 (instead of 321). In other words, the growth rate from 2008 to 2009 was relatively close to 0%.

But let's look at the overall benefit: in the twenty years since 1989 (the low point in the graph), the metric has increased by 418%. That's awesome!