# How to find mean relative differences?

I'm trying to describe mean differences between two populations $x_1$ and $x_2$, which are non-zero and positive. Their distribution is approximately beta with a positive skew. For example, with R:

# x1 <- round(rbeta(10, 1, 100)*1000, 1)
# x2 <- x1 + round(rbeta(10, 1, 100)*100, 1)
x1 <- c(1.7, 12.6, 22.3, 37.3, 15.2, 7.1, 31, 4.4, 9.5, 1.9)
x2 <- c(1.9, 14.2, 25.6, 39.2, 15.9, 8.7, 32.2, 7, 9.7, 1.9)


The two ways I can think of to determine the mean relative differences yield different numbers:

mean((x2 - x1)/x1)             # 0.1365628
(mean(x2) - mean(x1))/mean(x1) # 0.09300699


Why are they different, and which method is more descriptive of what I'm looking for? I.e., is $x_2$ 13.6% or 9.3% greater than $x_1$?

In the first formula, mean((x1 - x2)/x1) you are describing the differences between paired members of each group, (Wikipedia). Think of individuals before and after treatment.

In the second formula, (mean(x2) - mean(x1))/mean(x1), you are describing the differences between two, presumably independent, groups as a whole. Think of two independent samples.

I am not sure we can answer the question of what's better for you. If x1 is close to zero, then the first method would generate Cauchy normal variates, and if the population mean is zero, so would the second method. If both population means are zeroes, then whatever you do with them in terms of their ratios will end up as 0/0.