# Average of rate of change in a set of independent change events

What is the right method for computing average of percentage values? Arithmetic average, geometric average, or something else?

Here's an example of problems I'm talking about. Suppose for years 2009, 2010, 2011, the annual GDP growth rates are 1%, 2%, 3%, respectively. What is the average growth rate over these three years? I'm not talking about the domain of GDP growth in specific (it's just a made-up example). Instead, I want to know the way to compute average of percentiles in general (or at least somewhat smaller domain of average of certain growth rates).

* Update #1 - another example

Suppose I have percentages Pi=1 - Ni/Di (i=1, 2, ..., m). Ni is the execution time of program i after optimization, and Di is the execution time of the same program i before optimization. Then, Pi represents the rate of reduction in execution time after optimization. How can we compute the average rate of reduction in execution time for these m programs? The arithmetic average avg(Ni) is not meaningful, right?

* Update #2 - change of title

From "Average of percentage values" to "Average of rate of change in a set of independent change events"

Change event ei: change from original value ai to new value bi, and change rate pi=f(ai,bi)

What is the average change rate for a set of events {e1, e2, ..., en}?

• Taking the average of the percentages is probably not a very good idea. Do you know the numerator and the denominator counts of each of the percentages, or you just have the percentages, nothing else? I mean you can add up all the numerators (say, $N1+N2+N3$) and also add up all the denominators (say, $D1+D2+D3$) separately and then find the percentage as $\frac{(N1+N2+N3)}{(D1+D2+D3)}100\%$. – Blain Waan Nov 4 '12 at 10:31
• when you say What is the average growth rate over these three years? what you need is a type of average suitable for averaging growth rates, rather than for averaging percentages. Percentages occur in a variety of ways and different kinds of averages are used depending on what you're averaging. – Glen_b Nov 4 '12 at 10:56
• @BlainWaan You are right. That's one way I've tried before. But sometimes, the sum of N1, N2, N3 is not very meaningful (see my updated post). In those cases, I compute the geometric mean N of N1N2N3, and geometric mean D of D1D2D3, and use N/D as the average. Is that reasonable? – dacongy Nov 4 '12 at 12:02

## 1 Answer

To average growth rates the geometric mean seems appropriate.

$G = (\prod_{i=1}^n x_i)^\frac{1}{n}$

For example, suppose the rate is .10, .02 and .20. Then the total growth is $1.1*1.02*1.2$ over 3 years, or the cube root of that product per year.

$1.1*1.02*1.2 = 1.346$

and $1.346^\frac{1}{3} = 1.1104$

and, of course, cubing that gets back to 1.346.