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 normalized^{1},
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 folks^{2}, especially since
there are other transformations that behave similarly^{3}. 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
Paradox^{4} 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.