index.md (5651B)
1 --- 2 title: "Average percentage change is not the percentage change of the average" 3 date: 2024-08-11T19:42:59-07:00 4 draft: false 5 katex: true 6 categories: 7 - Science 8 - Data Science 9 tags: 10 - Statistics 11 --- 12 13 When I present data to non-technical stakeholders, I sometimes express 14 differences (typically changes over time) in terms of _percentage change_. 15 This indicator has the advantage of being unitless and normalized[^norm], 16 and is familiar to broad audiences (although some aspects are 17 counter-intuitive---see below). 18 19 The ratio of the difference between two observations (_x₀_ and _x₁_) and the 20 starting observation provides the _relative change_: 21 22 $$ \frac{x_1 - x_0}{x_0} $$ 23 24 Multiplying this proportion by 100 yields the percentage change. 25 26 Usually I have more than two observations separated by time---I’ll have _x₀_ 27 and _x₁_ values for many observational units. Consider the following set of 28 (made up) test scores: 29 30 | Student | Test 1 | Test 2 | Percentage Change | 31 |------------:|-------:|-------:|-------------------:| 32 | 1 | 75.0 | 81.0 | 8.00% | 33 | 2 | 80.0 | 82.0 | 2.50% | 34 | 3 | 80.0 | 86.0 | 7.50% | 35 | 4 | 96.0 | 90.0 | -6.25% | 36 | 5 | 100.0 | 90.0 | -10.00% | 37 | **Average** | 86.2 | 85.8 | 0.35% | 38 39 The average score on Test 2 was lower than that for Test 1, so one might 40 conclude that the students did worse. However, the average percentage change 41 of students’ scores is positive; in other words, relative to their starting 42 scores, students typically did better on the second test. This toy example 43 illustrates an important point: the average of the percentage change (0.35% 44 here) is not the same thing as the percentage change of the averages 45 (approximately -0.46% in this case). 46 47 I find the mathematical definition of these two measures illustrative. I’ll 48 stick with relative changes to keep the equations tidier. 49 50 The average of the relative change is: 51 52 $$ \mathbb{E}\[ \frac{x_1 - x_0}{x_0} \] = \mathbb{E}\[\frac{x_1}{x_0}\] - 1$$ 53 54 On the other hand, the relative change of the averages is: 55 56 $$ \frac{\mathbb{E}[x_1] - \mathbb{E}[x_0]}{\mathbb{E}[x_0]} = 57 \mathbb{E}[\frac{x_1}{\mathbb{E}[x_0]}] - 1$$ 58 59 One way to think about these equations is that the _average relative 60 change_ scales each _x₁_ by its associated _x₀_ before calculating the mean 61 of the scaled value. The _relative change of the averages_ scales each _x₁_ 62 by the _mean_ of _x₀_, and then takes the mean of these. 63 64 ## Which one to use 65 66 Since they measure different things, each measure can be appropriate in 67 different circumstances. 68 69 If _x₀_ and _x₁_ comprise a sample from a larger population, then the sample 70 mean of _x₀_ and _x₁_ are reliable estimates of the population mean of these 71 values. In this case, the percentage change of the averages is probably most 72 appropriate; it will estimate the percentage change in the population. 73 74 On the other hand, if the observational units represent an entire 75 population, or when using relative change to compare the behavior of 76 different measures over the same time period, then calculating 77 statistics---including the mean---of the relative changes is useful. 78 79 ## Why I’m sharing this 80 81 Perhaps this distinction is obvious to most folks[^folks], especially since 82 there are other transformations that behave similarly[^transform]. As 83 somebody accustomed to wanting the percentage change of the averages, 84 I recently saw---in real world data---a fairly large discrepancy between 85 that and the average percentage change, after deciding that I needed the 86 latter. Only after checking my code for obvious errors did I confirm that 87 the math made sense. The experience reminds me of encountering [Simpson’s 88 Paradox](https://en.wikipedia.org/wiki/Simpson%27s_paradox)[^simpson] in 89 real data. 90 91 ## Other measures of relative change 92 93 I avoid using percentage change as defined here (when I can) because it can 94 lead to confusion. Among percentage change’s problems is its lack of 95 “symmetry”; for example, an increase from 4 to 5 represents a 25% change, 96 while a decrease from 5 to 4 represents a -20% change. This is addressed by 97 other measures of relative change, such as the “arithmetic mean change” 98 (where the difference is scaled by the mean of _x₀_ and _x₁_ rather than 99 _x₀_ alone) or “logarithmic change” (where relative change is represented by 100 the natural logarithm of the ratio of _x₁_ to _x₀_). Note that even with 101 these measures, the average of relative changes are still distinct from the 102 relative change of the averages; however, a symmetrical measure is probably 103 a better choice when averaging changes. 104 105 ## Percentage change in practice 106 107 Since percentage change is only a _point estimate_, its presence should 108 supplement---and never replace---an appropriate model-based approach to 109 estimating the size of the change and the uncertainty of that estimate. 110 Still, percentage change is a useful measure to include in reports and slide 111 decks when it will be familiar to the audience, which is typically the case 112 in a business setting. I hope this helps somebody else select the right 113 calculation, and explain why the results are totally different from 114 a “wrong” one. 115 116 [^norm]: In the colloquial sense, not the statistical one. 117 118 [^folks]: At least the ones who’d read a post like this. 119 120 [^transform]: For example, the mean of the logarithm of a variable is not 121 the same as the logarithm of the mean of a variable. 122 123 [^simpson]: I’m not sure if this could be considered a special case of 124 Simpson’s paradox.