When I present data to non-technical stakeholders, I sometimes express differences (typically changes over time) in terms of percentage change. This indicator has the advantage of being unitless and normalized1, and is familiar to broad audiences (although some aspects are counter-intuitive—see below).
The ratio of the difference between two observations (x₀ and x₁) and the starting observation provides the relative change:
$$ \frac{x_1 - x_0}{x_0} $$
Multiplying this proportion by 100 yields the percentage change.
Usually I have more than two observations separated by time—I’ll have x₀ and x₁ values for many observational units. Consider the following set of (made up) test scores:
| Student | Test 1 | Test 2 | Percentage Change |
|---|---|---|---|
| 1 | 75.0 | 81.0 | 8.00% |
| 2 | 80.0 | 82.0 | 2.50% |
| 3 | 80.0 | 86.0 | 7.50% |
| 4 | 96.0 | 90.0 | -6.25% |
| 5 | 100.0 | 90.0 | -10.00% |
| Average | 86.2 | 85.8 | 0.35% |
The average score on Test 2 was lower than that for Test 1, so one might conclude that the students did worse. However, the average percentage change of students’ scores is positive; in other words, relative to their starting scores, students typically did better on the second test. This toy example illustrates an important point: the average of the percentage change (0.35% here) is not the same thing as the percentage change of the averages (approximately -0.46% in this case).
I find the mathematical definition of these two measures illustrative. I’ll stick with relative changes to keep the equations tidier.
The average of the relative change is:
$$ \mathbb{E}[ \frac{x_1 - x_0}{x_0} ] = \mathbb{E}[\frac{x_1}{x_0}] - 1$$
On the other hand, the relative change of the averages is:
$$ \frac{\mathbb{E}[x_1] - \mathbb{E}[x_0]}{\mathbb{E}[x_0]} = \mathbb{E}[\frac{x_1}{\mathbb{E}[x_0]}] - 1$$
One way to think about these equations is that the average relative change scales each x₁ by its associated x₀ before calculating the mean of the scaled value. The relative change of the averages scales each x₁ by the mean of x₀, and then takes the mean of these.
Which one to use
Since they measure different things, each measure can be appropriate in different circumstances.
If x₀ and x₁ comprise a sample from a larger population, then the sample mean of x₀ and x₁ are reliable estimates of the population mean of these values. In this case, the percentage change of the averages is probably most appropriate; it will estimate the percentage change in the population.
On the other hand, if the observational units represent an entire population, or when using relative change to compare the behavior of different measures over the same time period, then calculating statistics—including the mean—of the relative changes is useful.
Why I’m sharing this
Perhaps this distinction is obvious to most folks2, especially since there are other transformations that behave similarly3. As somebody accustomed to wanting the percentage change of the averages, I recently saw—in real world data—a fairly large discrepancy between that and the average percentage change, after deciding that I needed the latter. Only after checking my code for obvious errors did I confirm that the math made sense. The experience reminds me of encountering Simpson’s Paradox4 in real data.
Other measures of relative change
I avoid using percentage change as defined here (when I can) because it can lead to confusion. Among percentage change’s problems is its lack of “symmetry”; for example, an increase from 4 to 5 represents a 25% change, while a decrease from 5 to 4 represents a -20% change. This is addressed by other measures of relative change, such as the “arithmetic mean change” (where the difference is scaled by the mean of x₀ and x₁ rather than x₀ alone) or “logarithmic change” (where relative change is represented by the natural logarithm of the ratio of x₁ to x₀). Note that even with these measures, the average of relative changes are still distinct from the relative change of the averages; however, a symmetrical measure is probably a better choice when averaging changes.
Percentage change in practice
Since percentage change is only a point estimate, its presence should supplement—and never replace—an appropriate model-based approach to estimating the size of the change and the uncertainty of that estimate. Still, percentage change is a useful measure to include in reports and slide decks when it will be familiar to the audience, which is typically the case in a business setting. I hope this helps somebody else select the right calculation, and explain why the results are totally different from a “wrong” one.