How to compare percentage changes from groups with different sample sizes? I'm sure this is a simple problem. I had it years ago and was never able to figure out and to this day it still comes up as a random shower tought. I used to work in market planning, and kept track of the percentage change of sales by region from one year to the next. Some regions differ by magnitude on their size, as you can see on the table below. The smallest regions are obviously more "volatile", since very small increments in absolute units shifts the percent change dramatically.
My question is, is there some way to determine to "normalize" the percent changes or are there some statistical tests that could be used if the percent change of smaller regions are significantly bigger than those of larger regions?




Region
2020
2021
% Change




A
20
23
15%


B
250
275
10%


C
2000
2100
5%




If the underlying data reasonably approximate a Poisson distribution, an alternative to percent change is what I call a Poisson Z-score, calculated as:
$$2 \cdot ( \sqrt{\text{post}} - \sqrt{\text{pre}} ) = Z_p $$
And $Z_p$ follows approximately a standard normal distribution. I did this for monitoring low count crime data, and the approximation works reasonably well for counts even as low as a mean of 5 over time.
So here is what your table would look like:




Region
2020
2021
% Change
Zp




A
20
23
15%
0.64


B
250
275
10%
1.54


C
2000
2100
5%
2.21




My experience is for larger counts the Poisson assumption tends to be become less tenable, and so will reject more often than it should. In that case, if you have historical data over time, you could actually estimate the variance instead of relying on the Poisson assumption of mean=variance to calculate a difference over time.
