How can an average decrease in numbers show up as an average percentage increase?

Question

In the list below:

• the first column shows the population
• the second column shows the number by which the population has increased (so the total new population in the first row would be 27312).
• the third column shows the percentage increase i.e. column 2 / column 1

I then took the average of the second column and got an average number. But then I took the average of the third column and got an average percentage increase.

That doesn't make sense to me. My brain is a little fried. I'm trying to get an intuition for why this is.

• Surely an average fall in numbers should mean an average fall in percentage change?

Answer 1

Look at a two row example:

  1    9   1000%
100  -50    -50%

The numbers in second row are much larger than those in the first, so influence the average numerical change a lot. The percentages in the last column ignore the relative sizes of the rows.

The moral of the story is that averaging percent changes is almost always a bad idea.

Answer 2

When you average the percentages, you’re giving each of them the same weight in the average. But a $10\%$ change starting at $100,000$ is a change of $10,000$, while a $10\%$ change starting at $10,000$ is a change of only $1000$. That first change has $10$ times the effect on the overall population, even though it’s an identical percentage of its base population. If the first is an increase and the second a decrease, the average of the percentages as $0\%$, but the total population has gone from $110,000$ to $119,000$, an increase of $9000$, or $8.\overline{18}\%$.

For an average of the percentages to be meaningful, it would have to be a weighted average, with each percentage weighted by the fraction of the total population that it affects. In this case the first change affects $\frac{10}{11}$ of the total population of $110,000$, while the second affects only $\frac1{11}$ of it, so the weighted average of the percentages is

$$\frac{10}{11}(10)+\frac1{11}(-10)=\frac{90}{11}=8\frac2{11}=8.\overline{18}\,,$$

which is indeed the percentage by which the total population has changed.