A rose is a rose, but a trend rate is rarely what it seems.
What's in a Number?
Trends are a pattern of change over time. One way to express them is as an average rate. For example, you might hear, "Over the past three years, we've averaged 5% growth annually." Any given year in those three years is probably not 5%. For example, it was 2% for one year, followed by 7% and 6% last. You could show this as a geometric trend:
$$(1+0.02) \times (1+0.07) \times (1+0.06) \approx 1.16$$
Taken together, this averages to 5% each of the three years:
$$(1+0.05) \times (1+0.05) \times (1+0.05) \approx 1.16$$
Trend rates are nice because they're a single number. You can insert one in prose without the hassle of collecting multiple data points and creating a chart that needs real estate on your page. However, a single number hides a lot of context:
- How long of a time frame is this trend for?
- What period does this trend start?
- What's in the data set that informed this trend?
Trend Length
First, the length of the time frame matters. This mock time series data shows an expanding length and how the average trend at each length changes:
Any change in the number of periods used changes the average trend. For example, the average annual CPI over the past 20 years is 2.6%, but over the past 10 years it's 2.8% (Admittedly, not the most contrasting example).
Trend Period
Next, the starting period matters as well.
You might have the same length of time the trend applies for, but you can see how shifting the starting period changes the trend, too. For another example, a 5-year CPI trend starting in 1990 is 3.2%, but if you start in 2010, you get 1.8% on average over five years.
Trend Data
Last, even if you have the same length and starting period, you still need to understand what's in the underlying data. This point is more nuanced than the first two because one advantage of expressing trends as an average annual rate is that it is a relative change, allowing you to compare trends across related data. Be warned, the more different the underlying data, the more degraded the comparison.
For example, suppose you own some pharmacies. Pharmacy A is in a metro area and fills tens of thousands of prescriptions annually. Pharmacy B is in a more rural area and fills only a few thousand.
Scenario No. 1: You compare the average annual rate of prescriptions filled at pharmacies A and B over the same period and length of time. While the magnitudes of prescriptions differ (Pharmacy A has way more than Pharmacy B), the relative trends allow comparison.
Scenario No. 2: Repeat the prior scenario, but the comparison now includes both brand and generic drugs at Pharmacy A and only generic drugs at Pharmacy B. If brand drugs account for a noticeable share of prescriptions or exhibit notably different underlying trends, the comparison conflates the effects of the brand-versus-generic mix. From the outside, it would be hard to detect this confusion if you're only told something like, "The average annual growth in prescriptions filled over the past two years is 4% at Pharmacy A, and 6% at Pharmacy B." This statement doesn't clarify how prescriptions are counted by brand or generic. Even if you knew about the brand and generic differences, would you have enough context to estimate their consequences for the comparison? The two pharmacies could actually have near-identical trend rates for all prescriptions, but this fact is obscured if the underlying data differ.
Scenario No. 3: Repeat the first scenario, but this time it is the trend in prescription counts for Pharmacy A versus the prescription revenue for Pharmacy B. Being quoted two relative trends, you may not be given the underlying units, leading to an accidental and unequal comparison. This mismatch is a common danger when comparing across states or countries, and the underlying statistic is defined differently between locations or different measures are reported altogether. Moreover, it will be hard to tell, because sometimes you only get the one almighty standalone number.
Pause, Discern
In the future, when you see a trend rate with little context, treat it with circumspection, especially when compared to another rate. How was the underlying data defined—sampled? Over what period was this averaged? What period was used? How consistent are these answers between the compared rates? Remember, this will be tricky because you won't have an external cue to seek context if it's missing from the presented trend rate. Stay vigilant.
Calculations and graphics done in R version 4.3.3, with these packages:
Qiu Y (2024). showtext: Using Fonts More Easily in R Graphs. R package version 0.9-7. https://CRAN.R-project.org/package=showtext
Wickham H, et al. (2019). Welcome to the tidyverse. Journal of Open Source Software, 4 (43), 1686. https://doi.org/10.21105/joss.01686
Wilke C, Wiernik B (2022). ggtext: Improved Text Rendering Support for 'ggplot2'. R package version 0.1.2. https://CRAN.R-project.org/package=ggtext
Generative AIs like Anthropic's Claude Opus 4.5 were used in parts of coding and reviewing the writing. Cover art was created by the author with Midjourney and GIMP.
This website reflects the author's personal exploration of ideas and methods. The views expressed are solely their own and may not represent the policies or practices of any affiliated organizations, employers, or clients. Different perspectives, goals, or constraints within teams or organizations can lead to varying appropriate methods. The information provided is for general informational purposes only and should not be construed as legal, actuarial, or professional advice.
