Hockey Stick Growth

Graph — Fig 1. Variants of a “hockey stick curve”

Any business would love to generate “hockey stick growth.” Our tagline, “Deeper Insight, Steeper Growth” is intended to align to those aspirations. I used the hockey stick metaphor in a recent article, describing AI search trends as “a classic hockey stick curve that turns sharply upward starting in Q4 2022.” Seeing a friend’s recent post about hockey sticks, as in the physical version, led me to think about their shape and then to wonder if we all see the same curve in our mind when used to describe a trend. As an analytics consultant and practitioner who uses visualization, I was curious enough to investigate. Is there a “classic” hockey curve? Are there variations? Is the shape so well defined that Gen AI tools can generate that shape?

Through some searching and a few queries to generative AI engines, I discovered there are at least six curve shapes that could be interpreted as a hockey stick. Narrowing to the three that ended in an uptrend, I asked my three different AI engines to summarize common knowledge, gather examples and help me generate some visuals. Among my prompts was this one:

There seem to be three variants in common use, a j-type shape, a short flat followed by long steady incline, and a long flat, followed by a recent sharp uptick. Create a more formal explanation of these three, including an intro, labeling them in the same order but as A/B/C, and providing for each: a few citations (links), any famous associations, a visual graph, and an excel formula that could be used to portray it within a 100 wide (x axis), 1000 high (y axis) cartesian graph

I regularly use premium versions of generative AI engines provided by three of the leading platforms: Anthropic’s Claude Opus, Google’s Gemini, and Open AI’s ChatGPT. They each have relative strengths in weaknesses. For a summary of the common interpretations and industry associations of hockey curves, I liked Claude’s answer best. In addition to the except below, it provided several citations to business books and academic journals, all of which proved out as being real and relevant.

The term "hockey stick" curve is commonly used to describe graphs that exhibit a long period of relatively flat growth followed by a sharp, significant increase. In data visualization and various business and scientific contexts, three primary variations of the hockey stick curve are frequently observed: the J-curve, the gradually increasing curve, and the classic hockey stick curve.

A. J-curve:
The J-curve is characterized by an initial decline followed by a rapid, sharp increase, resembling the letter "J". This curve is often associated with economic trends, such as the devaluation of a currency or the impact of a trade policy change on a country's trade balance.

Famous associations:
- The J-curve effect in economics (Bahmani-Oskooee & Ratha, 2004)
- The J-curve in private equity fund returns (Phalippou & Gottschalg, 2009)

B. Gradually increasing curve:
This variation of the hockey stick curve features a short period of flat growth followed by a long, steady incline. This curve is common in contexts such as product adoption or market growth.

Famous associations:
- The diffusion of innovations theory (Rogers, 2003)
- The technology adoption lifecycle (Moore, 1991)

C. Classic hockey stick curve:
The classic hockey stick curve is characterized by a long period of flat growth followed by a recent, sharp uptick. This curve is often seen in contexts such as startup growth, environmental data, or technological progress.

Famous associations:
- The hockey stick graph in climate change (Mann, Bradley, & Hughes, 1999)
- The hockey stick growth of successful startups (Bussgang, 2010)

In conclusion, these three variations of the hockey stick curve - the J-curve, the gradually increasing curve, and the classic hockey stick curve - are widely used in data visualization to represent various patterns of growth and change across different domains, from economics and business to environmental science and technology.

Image of Cartesian graph — Fig 2. Three variations of “hockey stick” curve as generated by Open AI’s DALL-E (May 2024)

As for the visuals and formulas I requested, none of the generative AIs produced something that met my needs. The results ranged from ineffective to comical, despite multiple prompts to revise or take fresh approaches. It seems the current state of LLMs are better at analyzing what’s already published than generating something new, even with the best attempts at a clear description. I use AI a lot for Excel formulas. The models are usually accurate on the first try. In this case, they all struggled to generate curves that fit within a set range, making it hard to show a comparison of the graphs. The visuals generated weren’t much better. Google’s Gemini managed to create some basic Cartesian plots, but they weren’t sufficient to show a comparison of the three curve types. Claude is becoming my go-to AI platform, but it’s not yet able to generate images. As for ChatGPT and the DALL-E capabilities you get with their premium model, what’s good for producing creative images doesn’t (yet) do so well for generating meaningful data visualization. See Fig 2.

The goal of generating some comparable examples of the three varieties was to explain and to see what others thought. I ended up using the curve shape tool in PowerPoint to create the visual I used in a poll on LinkedIn. See Fig 1. Though the response rate was low, my LinkedIn connections tend to work in the intersection of business and data visualization. The majority picked variant C, “a long period of flat growth followed by a recent, sharp uptick” as most closely matching what they envision when they think of a hockey stick curve. That shape also matches the AI search trends I had referenced as the “classic.” I note that Claude’s analysis also labeled that form as “classic,” without my ever including that specific word in my prompts.

Data visualization manifests the old adage that a “picture is worth a thousand words.” As analysts, we still need words to describe trends. Visual metaphors are helpful. However, we need to realize that they may convey different images based on the audience, their professional background and their personal experiences.

Insights to Catalyze Your Success

A “hockey stick curve” does typically signal steeper growth, but with meaningful variations in the types of patterns leading up to the turning point.
In their current state, generative AI engines are effective at analyzing common trend patterns, but they struggle to re-create the shapes accurately and for relevant comparison.
Metaphors can be helpful for communicating analytic concepts. Make sure to consider the audience perspective. When possible, include a visual.