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


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.
