# Weighted average of percentage increase

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; and 5,000,000 respectively.

Hence, the percentage 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? Maybe a standard average is sufficient?

Thanks very much for your inputs.

You have 1, 4.5, 5 in successive years and, as you noted, the growth in the first two years is 350% and in the second two years, only 11.1%; again as you noted, the arithmetic mean of these (expressed as proportions) is $\frac{3.5 + .111}{2} = 1.805$. Yet that is not the average growth: $1*1.805*1.805 = 3.205$ but your final value is 5.
You could express these as ratios of 4.5 and 1.11, then the average is 2.8. But $2.8*2.8 = 7.84$, again, not 5.
What you want is the geometric mean of the ratios: $(4.5*1.11)^{0.5} = 2.23$ and, indeed, $2.23*2.23 = 5$ (except for rounding error).