# Percent change over time

I have the following data:

$$\begin{array}{c|c|c}2010 & 2011 & 2012 \\\hline 50 & 20 & 30 \\\end{array}$$

How would the percent change be calculated?

Like this: $$100\times\left(\frac{|30-50|}{50}\right)\quad?$$

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It depends on whether you want the absolute percent change (i.e., a decrease still produces a positive number), or the relative percent change (i.e. a decrease produces a negative number). It also depends on which times you are looking at.

So, to compute the absolute percent change from 2010 to 2012, your formula is correct, and we get $$100\times\left(\frac{|30-50|}{50}\right)=100\times\left(\frac{20}{50}\right)=40\%.$$

But to compute the relative percent change from 2010 to 2012, we drop the absolute value from the formula, and get $$100\times\left(\frac{30-50}{50}\right)=100\times\left(\frac{-20}{50}\right)=-40\%.$$

To compute the relative percent change from time $A$ to time $B$, take the data points at $A$ and $B$ (let's call them $f(A)$ and $f(B)$, respectively), and compute $$100\times\left(\frac{f(B)-f(A)}{f(A)}\right)$$ To compute the absolute percent change, you'd take the absolute value.

In this problem, I believe it is implied that the times you should be comparing are the first and the last, but a different problem might ask you about comparing different times, in which case you'll have to be careful to use the data points corresponding to those times.

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Just curious, why are the intermediate points not taken into account? –  O.O Feb 9 '12 at 23:32
When we're doing these calculations, all we're looking at are the two moments in time, time $A$ and time $B$, and comparing the data we have at those times. A more in-depth analysis of a collection of data might look at a "moving average" or other measurements that do take the intermediate points into account. –  Zev Chonoles Feb 9 '12 at 23:35
@ZevChonoles For clarfication, would you describe this as a 3-year percent change (2010-2012) or a 2-year change (2010-11, 2011-12, since there's 2 intervals between 3 years)? –  David0044 Jun 10 '13 at 17:28
@David: It is a change over the course of two years. In other words, the intervals themselves are the years, not the calendar dates. Jan 1, 2010 to Jan 1, 2011 is one year, and Jan 1, 2011 to Jan 1, 2012 is another year. Of course, if the data is not collected at the same time each year, then one should use a more precise description, with months or weeks (e.g., the time period from Jan 1, 2010 to Dec 31, 2011 is almost two years, so calling it one year would be misleading, calling it two years would be fine, and calling it one year, 51 weeks and 6 days would be exact). –  Zev Chonoles Jun 10 '13 at 17:34