45 Ways to Communicate Two Quantities

Back in 2010, I was giving a workshop on interactive data visualization in Lima, Perú, discussing whether a dataset has a unique or at least an ideal way to be visualized. For a simple data structure — a list of some hundreds of numbers, for instance — around half of 20 participants were convinced that there’s one way that is clearly better in communicating the data, regardless of the unit of the values, their range, meaning, context and possible aim of the visualization. This discussion actually came out as a consequence of another idea, which resonated with most participants, as well: that there should be a guide that indicates the best way to visualize each possible dataset. So I proposed the following exercise:

let’s try to find all possible ways to visualize a ludicrously small data set of two numbers. Afterwords, let’s try to pick the best visualization.

With such a tiny dataset, you would think we would complete both exercises in less than 5 minutes. Yet, we spent more than two hours without having actually accomplished either of the two tasks. Not only was the number of possible ways to visualize two values far higher than expected, but also each single visualization method admitted multiple and interesting variations and opened new questions and discussions. The following is a list of different ways to visualize two numbers. In many cases, the visualization depends directly on the unit, meaning, interpretation and context of the values. Some of the examples might be seen as variations of the same theme, others will be perceived as excessively eccentric or even esoteric. On the other hand, I’m sure many of you will come up with visualization methods not included in the list (in which case, please include them in the comments!). I explain only some of the methods; in general they are clear and well known. Some have a very bad reputation, but I won’t get into that kind of discussion. I’ll also avoid the discussion about which method is better according to the context and goal.

75 and 37

1. writing, number notation   2. squares   3. repeated icon   4. hundreds, tens, units, decimals… represented by squares   5. bars   6. line graph   7. percentages bars   8. spliced bar   9. proportion

For the case one value is a part of the other.
  10. interval   11. squares merged   12. percentages in squares   13. pie charts   14. donut chart
It differs from the pie chart because it can be nested into other donut charts in order to create a more complex visualization.
  15. circle areas   16. semi-circle areas
Can be placed on a bi-dimensional space associating the two values to a single point.
  17. circle and external ring
The (internal) circle surface is proportional to the lower value, and the (external) ring surface to the higher one. [!] It might be interpreted that the second value is represented by the complete bigger circle, including the hidden surface. I’ve made some tests on this, asking people, and obtaining different interpretations.
  18. co-centered circles
The lower value is represented by the internal surface, and the higher value by the external circle surface, including its non visible part (that’s hidden by the smaller circle). [!] Can be interpreted as the case 17. Adding a shadow or even some tridimensional representation are attempts to ‘separate’ the two circles, thus trying to avoid the ambiguity.
  19. square divided
[!] Same ambiguity problem as in 17. and 18.: the reader can interpret that the bigger number is associated with the entire square, or only with the visible part.
  20. shape divided   21. square surfaces   22. shape surfaces   23. different shape surfaces   24. icon surfaces   25. icon height
[!] In this case, it is the metaphor and an explicit message making it clear that the values are represented by the height of the icon and not its surface.
  26. volumes
In a the volume is proportional to the object’s height, not the case for b, which is very difficult to read.
  27. special metaphors
…for special units: a gauge for speeds, balances for weights
  28. gray tones   29. color scale   30. geometric proportions   31. horizontal/vertical proportions
Useful if the first value represents a horizontal distance, and the second value a vertical one.
  32. coordinates
a cartesian and b polar
  33. angles
In this case the angle's unit is grades.
  34. geographic coordinates
a longitude, latitude b latitude, longitude
  35. values associated to countries   36. density   37. percentages / density
Combination of 12 and 36.
  38. dashed
similar to 28 gray tones and 36 density.
  39. nodes and connections in a network
As we don’t have more than the quantitative information, this visualization is only a way to have a first glance of the external structure of the network (the relations are chosen randomly).
  40. parameters of a mathematical function   41. harmonic frequencies   42. pulses frequencies (in pulses per minute)   43. rotation frequencies (in revolutions per minute)   44. sound frequencies (in Herz) a. 75:[haiku url=”https://s3.amazonaws.com/scribblelive-com-prod/wp-content/uploads/2012/07/s75.wav”]     and 37:[haiku url=”https://s3.amazonaws.com/scribblelive-com-prod/wp-content/uploads/2012/07/s37.wav”] b. 37 to 75:[haiku url=”https://s3.amazonaws.com/scribblelive-com-prod/wp-content/uploads/2012/07/s37to75.wav”]   45. fat fonts

So what’s the point of this list?

I’m sure this list is not complete and could be organized in better ways. But I don’t think it’s possible to build a definitive list — it just doesn’t exist. New ideas will always appear (like in 45.) Among the methods mentioned in this list, some are scalable to bigger lists of numbers, and others are not. That’s because they don’t depict each value separately; rather, they say something about the relation between the values (for instance the ones that represent proportion, or the coordinate systems). By adding more numbers to the list, the amount of possible combinations rises exponentially. What happens if we move to other, richer datasets — lets say a table containing a list of countries, lists of numbers, and lists of dates? (I’m talking about small datasets anyway, nothing that could be called ‘big data’.) The data set not only contains more values, it contains different units. And not only could each list be interpreted and contextualized in different ways, the relations between lists and individual values can be interpreted and contextualized differently. This is the classic complexity context: simple elements that interact; you add more elements and the number of possible interesting (meaningful) patterns increases dramatically.

Information visualization is a language.

A language able to explain the world, tell stories, point specific facts and objects, elaborate ambiguous messages, defend arguments, attack arguments and carry ideas and ideologies. As any language, visualization works with combinatoric and generative rules. And as with any language, it’s just impossible to delineate its borders or enlist all the objects it can produce. It’s not a coincidence that I started and ended the list with text (from millennial to to the most recent notations): I see visualization as an extension of writing. By combining some of the methods of this list, it’s possible to create new ones: co-centered donut charts (14.) can be used to visualize tree structures, dashed shapes (38.) can be used on maps, line graphs with solid shapes (6.c.) can evolve to steam graphs, free shapes with surfaces proportional to values (23.) can be found in cartograms or non-rectangular treemaps; with fat fonts (45.) it’s possible to build special heat-maps, etc. The purpose of this list is to stimulate this kind of combinatorial, generative and creative thinking. Doing this exercise in Lima was a thoughtful experience for every participant, it raised multiple new ideas and debates, and that’s what I wanted to bring here. Don’t take this list as 45 answers to a question, but as 45 points of departure for new questions.


Exercices in Style FatFonts The Grammar of Graphics Information Graphics

A PDF poster of the images in this post can be downloaded here.   Santiago Ortiz is a mathematician, educator, developer, researcher , freelancer. He creates interactive visualization projects. http://moebio.com @moebio

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