I believe I have quite a simple problem, but want clarification on whether it is the best method to use.
Say I have have insurance line, and the net income (£) from this business for 2011, 2012, 2013 is
1,000,000
4,500,000
5,000,000
respectively.
Hence, the pecentage increase from 11-12 is 350%, and from 12-13 is 11.1%.
Now, the average of these is 180.5%, which is a representation of the average percentage increase across all years, but it has been suggested to me to instead represent the data by a "weighted" average.
I am curious... would the weights be the difference in income from each year?
i.e. Weighted Avg. = $((3.5)*350\% + (0.5)*11.1\%)/4 = 307\%$
or would the weights possibly the values themselves? Or maybe there is another option?
Thanks very much for your inputs.
Here is how you would usually compute the per annum increase: $$y_t=r^t y_0 \Leftrightarrow r=\left(\frac{y_t}{y_0}\right)^{1/t},$$ where $r-1$ is the increase per annum, $t$ is the amount of time periods (years) and $y_0$ and $y_t$ is in your case the insurance line in the first year and the last year. In your example, $r=\sqrt{5}\approx 2.23$, and the increase per annum is about $120\%$
The advantage of computing like this is that when you actually use $r$, then $y_0*r^t$, by construction, actually gives you the right end value $y_t$. Using your estimate of the annual increase, on the other hand, you get $y_0*(1+1.8)^2\approx 7,840,000$, which is far from $y_2=5,000,000$. Taking any linear average (weighted or unweighted) of a variable that is applied exponentially doesn't make much sense.