Pie charts and self-sufficiency

This graphic shows up in a recent issue of Princeton alumni magazine, which has a series of pie charts.

The story being depicted is clear: the school has been generously increasing the amount of financial aid given to students since 1998. The proportion receiving any aid went from 43% to 67% so about two out of three students who enrolled in 2023 are getting aid.

The key components of the story are the values in 1998 and 2023, and the growth trend over this period.

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Here is an exercise worth doing. Think about how you figured out the story components.

Is it this?

Or is it this?

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This is what I've been calling a "self-sufficiency test" (link). How much work are the visual elements doing in conveying the graph's message to you? If the visual elements aren't doing much, then the designer hasn't taken advantage of the visual medium.

Excess delay

The hot topic in New York at the moment is congestion pricing for vehicles entering Manhattan, which is set to debut during the month of June. I found this chart (link) that purports to prove the effectiveness of London's similar scheme introduced a while back.

This is a case of the visual fighting against the data. The visual feels very busy and yet the story lying beneath the data isn't that complex.

This chart was probably designed to accompany some text which isn't available free from that link so I haven't seen it. The reader's expectation is to compare the periods before and after the introduction of congestion charges. But even the task of figuring out the pre- and post-period is taking more time than necessary. In particular, "WEZ" is not defined. (I looked this up, it's "Western Extension Zone" so presumably they expanded the area in which charges were applied when the travel rates went back to pre-charging levels.)

The one element of the graphic that raises eyebrows is the legend which screams to be read.

Why are there four colors for two items? The legend is not self-sufficient. The reader has to look at the chart itself and realize that purple is the pre-charging period while green (and blue) is the post-charging period (ignoring the distinction between CCZ and WEZ).

While we are solving this puzzle, we also notice that the bottom two colors are used to represent an unchanging quantity - which is the definition of "no congestion". This no-congestion travel rate is a constant throughout the chart and yet a lot of ink of two colors have been spilled on it. The real story is in the excess delay, which the congestion charging scheme was supposed to reduce.

The excess on the chart isn't harmless. The excess delay on the roads has been transferred to the chart reader. It actually distracts from the story the analyst is wanting to tell. Presumably, the story is that the excess delays dropped quite a bit after congestion charging was introduced. About four years later, the travel rates had creeped back to pre-charging levels, whereupon the authorities responded by extending the charging zone to WEZ (which as of the time of the chart, wasn't apparently bringing the travel rate down.)

Instead of that story, the excess of the chart makes me wonder... the roads are still highly congested with travel rates far above the level required to achieve no congestion, even after the charging scheme was introduced.

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I started removing some of the excess from the chart. Here's the first cut:

This is better but it is still very busy. One problem is the choice of columns, even though the data are found strictly on the top of each column. (Besides, when I chop off the unchanging sections of the columns, I created a start-not-from-zero problem.) Also, the labeling of the months leaves much to be desired, there are too many grid lines, etc.

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Here is the version I landed on. Instead of columns, I use lines. When lines are used, there is no need for month labels since we can assume a reader knows the structure of months within a year.

A priniciple I hold dear is not to have legends unless it is absolutely required. In this case, there is no need to have a legend. I also brought back the notion of a uncongested travel speed, with a single line (and annotation).

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The chart raises several questions about the underlying analysis. I'd interested in learning more about "moving car observer surveys". What are those? Are they reliable?

Further, for evidence of efficacy, I think the pre-charging period must be expanded to multiple years. Was 2002 a particularly bad year?

Thirdly, assuming WEZ indicates the expansion of the program to a new geographical area, I'm not sure whether the data prior to its introduction represents the travel rate that includes the WEZ (despite no charging) or excludes it. Arguments can be made for each case so the key from a dataviz perspective is to clarify what was actually done.

 

P.S. [6-6-24] On the day I posted this, NY State Governer decided to cancel the congestion pricing scheme that was set to start at the end of June.

Aligning V and Q by way of D

In the Trifecta Checkup (link), there is a green arrow between the Q (question) and V (visual) corners, indicating that they should align. This post illustrates what I mean by that.

I saw the following chart in a Washington Post article comparing dairy milk and plant-based "milks".

The article contains a whole series of charts. The one shown here focuses on vitamins.

The red color screams at the reader. At first, it appears to suggest that dairy milk is a standout on all four categories of vitamins. But that's not what the data say.

Let's take a look at the chart form: it's a grid of four plots, each containing one square for each of four types of "milk". The data are encoded in the areas of the squares. The red and green colors represent category labels and do not reflect data values.

Whenever we make bubble plots (the closest relative of these square plots), we have to solve a scale problem. What is the relationship between the scales of the four plots?

I noticed the largest square is the same size across all four plots. So, the size of each square is made relative to the maximum value in each plot, which is assigned a fixed size. In effect, the data encoding scheme is that the areas of the squares show the index values relative to the group maximum of each vitamin category. So, soy milk has 72% as much potassium as dairy milk while oat and almond milks have roughly 45% as much as dairy.

The same encoding scheme is applied also to riboflavin. Oat milk has the most riboflavin, so its square is the largest. Soy milk is 80% of oat, while dairy has 60% of oat.

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Let's step back to the Trifecta Checkup (link). What's the question being asked in this chart? We're interested in the amount of vitamins found in plant-based milk relative to dairy milk. We're less interested in which type of "milk" has the highest amount of a particular vitamin.

Thus, I'd prefer the indexing tied to the amount found in dairy milk, rather than the maximum value in each category. The following set of column charts show this encoding:

I changed the color coding so that blue columns represent higher amounts than dairy while yellow represent lower.

From the column chart, we find that plant-based "milks" contain significantly less potassium and phosphorus than dairy milk while oat and soy "milks" contain more riboflavin than dairy. Almond "milk" has negligible amounts of riboflavin and phosphorus. There is vritually no difference between the four "milk" types in providing vitamin D.

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In the above redo, I strengthen the alignment of the Q and V corners. This is accomplished by making a stop at the D corner: I change how the raw data are transformed into index values. 

Just for comparison, if I only change the indexing strategy but retain the square plot chart form, the revised chart looks like this:

The four squares showing dairy on this version have the same size. Readers can evaluate the relative sizes of the other "milk" types.

Neither the forest nor the trees

On the NYT's twitter feed, they featured an article titled "These Seven Tech Stocks are Driving the Market". The first sentence of the article reads: "The S&P 500 is at an all-time high, and investors have just a handful of stocks to thank for it."

Without having seen any data, I'd surmise from that line that (a) the S&P 500 index has gone up recently, and (b) most if not all of the gain in the index can be attributed to gains in the tech stocks mentioned in the headline. (For purists, a handful is five, not seven.)

The chart accompanying the tweet is a treemap:

The treemap is possibly the most overhyped chart type of the modern era. Its use here is tangential to the story of surging market value. That's because the treemap presents a snapshot of the composition of the index, but contains nothing about the trend (change over time) of the average index value or of its components.

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Even in representing composition, the treemap is inferior to, gasp, a pie chart. Of course, we can only use a pie chart for small numbers of components. The following illustration takes the data from the NYT chart on the Magnificent Seven tech stocks, and compares a treemap versus a pie chart side by side:

The reason why the treemap is worse is that both the width and the height of the boxes are changing while only the radius (or angle) of the pie slices is varying. (Not saying use a pie chart, just saying the treemap is worse.)

There is a reason why the designer appended data labels to each of the seven boxes. The effect of not having those labels is readily felt when our eyes reach the next set of stocks – which carry company names but not their market values. What is the market value of Berkshire Hathaway?

Even more so, what proportion of the total is the market value of Berkshire Hathaway? Indeed, if the designer did not write down 29%, it would take a bit of work to figure out the aggregate value of yellow boxes relative to the entire box!

This design sucessfully draws our attention to the structural importance of various components of the whole. There are three layers - the yellow boxes (Magnificent Seven), the gray boxes with company names, and the other gray boxes. I also like how they positioned the text on the right column.

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Going inside the NYT article itself, we find two line charts that convey the story as told.

Here's the first one:

They are comparing the most recent stock prices with those from October 12 2022, which is identified as the previous "low". (I'm actually confused by how the most recent "low" is defined, but that's a different subject.)

This chart carries a lot of good information, even though it does not plot "all the data", as in each of the 500 S&P components individually. Over the period under analysis, the average index value has gone up about 35% while the Magnificent Seven's value have skyrocketed by 65% in aggregate. The latter accounted for 30% of the total value at the most recent time point.

If we set the S&P 500 index value in 2024 as 100, then the M7 value in 2024 is 30. After unwinding the 65% growth, the M7 value in October 2022 was 18; the S&P 500 in October 2022 was 74. Thus, the weight of M7 was 24% (18/74) in October 2022, compared to 30% now. Consequently, the weight of the other 473 stocks declined from 76% to 70%.

This isn't even the full story because most of the action within the M7 is in Nvidia, the stock most tightly associated with the current AI hype, as shown in the other line chart.

Nvidia's value jumped by 430% in that time window. From the treemap, the total current value of M7 is $12.3 b while Nvidia's value is $1.4 b, thus Nvidia is 11.4% of M7 currently. Since M7 is 29% of the total S&P 500, Nvidia is 11.4%*29% = 3% of the S&P. Thus, in 2024, against 100 for the S&P, Nvidia's share is 3. After unwinding the 430% growth, Nvidia's share in October 2022 was 0.6, about 0.8% of 74. Its weight tripled during this period of time.

Two metrics in-fighting

The Wall Street Journal shows the following chart which pits two metrics against each other:

The primary metric is the change in median yearly salary between the two periods of time. We presume it's primary because of its presence in the chart title, and the blue bars being more readable than the green bubbles. The secondary metric is the median yearly salary in the later period.

That, I believe, was the intended design. When I saw this chart, my eyes went to the numbers inside the green bubbles. Perhaps it's because I didn't read the chart title first, and the horizontal axis wasn't labelled so it wasn't obvious what the blue bars coded.

As with most bubble charts, the data labels exist to cover up the inadequacy of circular areas. The self-sufficiency test - removing the data labels - shows this well:

It's simply impossible to know what values should be in each bubble, or to perceive the relative sizes of those bubbles.

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Reversing the order of the blue bars also helps:

The original order is one of the more annoying features in most visualization packages. Because internally, the categories are numbered 1, 2, 3, ..., and because the convention is to have values run higher as they run up the vertical axis, these packages would place the top-ranked item at the bottom of the chart.

Most people read top to bottom, which means that they read the least important item first, and the most important item last!

In most visualization packages, it takes only 1 click or 1 action to reverse the order of the items. Please do it!

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For change over time, I like using a Bumps chart, otherwise called a slope graph:

An elaborate data vessel

I recently came across the following dataviz showing global oil production (link).

This is an ambitious graphic that addresses several questions of composition.

The raw data show the amount of production by country adding up to the global total. The countries are then grouped by region. Further, the graph presents an oil-and-gas specific grouping, as indicated by the legend shown just below the chart title. This grouping is indicated by the color of the circumference of the circle containing the flag of the country.

This chart form is popular in modern online graphics programs. It is like an elaborate data vessel. Because the countries are lined up around the barrel, a space has been created on three sides to admit labels and text annotations. This is a strength of this chart form.

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The chart conveys little information about the underlying data. Each country is given a unique odd shaped polygon, making it impossible to compare sizes. It’s definitely possible to pick out U.S., Russia, Saudi Arabia as the top producers. But in presenting the ranks of the data, this chart form pales in comparison to a straightforward data table, or a bar chart. The less said about presenting values, the better.

Indeed, our self-sufficiency test exposes the inability of these polygons to convey the data. This is precisely why almost all values of the dataset are present on the chart.

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The dataviz subtly presumes some knowledge on the part of the readers.

The regions are not directly labeled. The readers must know that Saudi Arabia is in the Middle East, U.S. is part of North America, etc. Admittedly this is not a big ask, but it is an ask.

It is also assumed that readers know their flags, especially those of smaller countries. Some of the small polygons have no space left for country names and they are labeled with just flags.

In addition, knowing country acronyms is required for smaller countries as well. For example, in Africa, we find AGO, COG and GAB.

For this chart form the designer treats each country according to the space it has on the chart (except those countries that found themselves on the edges of the barrel). Font sizes, icons, labels, acronyms, data labels, etc. vary.

The readers are assumed to know the significance of OPEC and OPEC+. This grouping is given second fiddle, and can be found via the color of the circumference of the flag icons.

I’d have not assigned a color to the non-OPEC countries, and just use the yellow and blue for OPEC and OPEC+. This is a little edit but makes the search for the edges more efficient.

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Let’s now return to the perception of composition.

In exactly the same manner as individual countries, the larger regions are represented by polygons that have arbitrary shapes. One can strain to compile the rank order of regions but it’s impossible to compare the relative values of production across regions. Perhaps this explains the presence of another chart at the bottom that addresses this regional comparison.

The situation is worse for the OPEC/OPEC+ grouping. Now, the readers must find all flag icons with edges of a specific color, then mentally piece together these arbitrarily shaped polygons, then realizing that they won’t fit together nicely, and so must now mentally morph the shapes in an area-preserving manner, in order to complete this puzzle.

This is why I said earlier this is an elaborate data vessel. It’s nice to look at but it doesn’t convey information about composition as readers might expect it to.

What is the question is the question

I picked up a Fortune magazine while traveling, and saw this bag of bubbles chart.

This chart is visually appealing, that must be said. Each circle represents the reported revenues of a corporation that belongs to the “Global 500 Companies” list. It is labeled by the location of the company’s headquarters. The largest bubble shows Beijing, the capital of China, indicating that companies based in Beijing count $6 trillion dollars of revenues amongst them. The color of the bubbles show large geographical units; the red bubbles are cities in Greater China.

I appreciate a couple of the design decisions. The chart title and legend are placed on the top, making it easy to find one’s bearing – effective while non-intrusive. The labeling signals a layering: the first and biggest group have icons; the second biggest group has both name and value inside the bubbles; the third group has values inside the bubbles but names outside; the smallest group contains no labels.

Note the judgement call the designer made. For cities that readers might not be familiar with, a country name (typically abbreviated) is added. This is a tough call since mileage varies.

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As I discussed before (link), the bag of bubbles does not elevate comprehension. Just try answering any of the following questions, which any of us may have, using just the bag of bubbles:

• What proportion of the total revenues are found in Beijing?
• What proportion of the total revenues are found in Greater China?
• What are the top 5 cities in Greater China?
• What are the ranks of the six regions?

If we apply the self-sufficiency test and remove all the value labels, it’s even harder to figure out what’s what.

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Moving to the D corner of the Trifecta Checkup, we aren’t sure how to interpret this dataset. It’s unclear if these companies derive most of their revenues locally, or internationally. A company headquartered in Washington D.C. may earn most of its revenues in other places. Even if Beijing-based companies serve mostly Chinese customers, only a minority of revenues would be directly drawn from Beijing. Some U.S. corporations may choose its headquarters based on tax considerations. It’s a bit misleading to assign all revenues to one city.

As we explore this further, it becomes clear that the designer must establish a target – a strong idea of what question s/he wants to address. The Fortune piece comes with a paragraph. It appears that an important story is the spatial dispersion of corporate revenues in different countries. They point out that U.S. corporate HQs are more distributed geographically than Chinese corporate HQs, which tend to be found in the key cities.

There is a disconnect between the Question and the Data used to create the visualization. There is also a disconnect between the Question and the Visual display.

Losing the plot while stacking up the bars

I came across this chart from an infographics that claims to show which zip codes in the U.S. are the "dirtiest" (link). I won't go into the data analysis in this post - it's the usual "open data" style analysis that takes whatever data they could find (in this case, 311 calls) and make some hay out of it.

It's amazing how such analyses frequently land on the Top N, Bottom N table. Top/Bottom N is euphemistically called "insights". But "insights" should answer at least one of these following questions: Where are these zip codes? What's the reason why 11216 has the highest rate of complaints while 11040 has the lowest? What measures can be taken to make the city cleaner?

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The basic form chosen for this graphic is the bar chart. The data concerns the number of complaints per 100,000 people (about sanitation - they didn't disclose how they classified a complaint as about sanitation).

To mitigate the "boredom" of bar charts, the designer made the edges of the bars swiggly, and added icons of items found in trash inside the bars. These are thankfully not too intrusive.

Why are all the data printed on the chart? Try mentally wiping the data labels, and you'll understand why the designer did it.

If readers look at data labels rather than the bars, then the data visualization surely has failed. I'd prefer to use an axis. 

If you spend a few more minutes on the chart, you may notice the gray parts. This is not the simple bar chart but a stacked bar chart. In effect, every bar is referenced to the first bar, which shows the maximum number of complaints per 100K people. For example, zip code 10474 has about 90% of the complaints experienced in zip code 11216, the "dirtiest" place in New York.

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The infographic then moves on to Los Angeles, and repeats the Top N/Bottom N presentation:

With this, the plot is lost.

For an inexplicable reason, the dirtiest zip code in LA does not occupy the entire length of the bar. The worst zip code here fills out 87% of the bar length, implying that the entire bar represents the value of 34,978 complaints per 100K people. How did the designer decide on this number?

As a result, every other value is referenced to 34,978 and not to the rate of complaints in the dirtiest zip code!

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The infographic eventually covers Houston. Here are the dirtiest two zip codes in Houston:

How does one interpret the orange section of the second bar? The original intention is for us to see that this zip code is about 80% as dirty as the dirtiest zip code. However, the full length of the bar does not here represent the dirtiest zip code.

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We also got a hint as to why this entire analysis is problematic. The values in LA are way bigger than those in NY, about 4 times higher at the top of the table. Is LA really that much dirtier than NY? Or perhaps the data have not been properly aligned between cities?

 

P.S. [8-26-2023] Added link to the infographic.

 

One bubble is a tragedy, and a bag of bubbles is...

From Kathleen Tyson's twitter account, I came across a graphic showing the destinations of Ukraine's grain exports since 2022 under the auspices of a UN deal. This graphic, made by AFP, uses one of the chart forms that baffle me - the bag of bubbles.

The first trouble with a bag of bubbles is the single bubble. The human brain is just not fit for comparing bubble sizes. The self-sufficiency test is my favorite device for demonstrating this weakness. The following is the European section of the above chart, with the data labels removed.

How much bigger is Spain than the Netherlands? What's the difference between Italy and the Netherlands? The answers don't come easily to mind. (The Netherlands is about 40% the size of Spain, and Italy is about 20% larger than the Netherlands.)

While comparing relative circular areas is a struggle, figuring out the relative ranks is not. Sure, it gets tougher with small differences (Germany vs S. Korea, Belgium vs Portugal) but saying those pairs are tied isn't a tragedy.

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Another issue with bubble charts is how difficult it is to assess absolute values. A circle on its own has no reference point. The designer needs to add data labels or a legend. Adding data labels is an act of giving up. The data labels become the primary instrument for communicating the data, not the visual construct. Adding one data label is not enough, as the following shows:

Being told that Spain's value is 4.1 does little to help estimate the values for the non-labelled bubbles.

The chart does come with the following legend:

For this legend to work, the sample bubble sizes should span the range of the data. Notice that it's difficult to extrapolate from the size of the 1-million-ton bubble to 2-million, 4-million, etc. The analogy is a column chart in which the vertical axis does not extend through the full range of the dataset.

The designer totally gets this. The chart therefore contains both selected data labels and the partial legend. Every bubble larger than 1 million tons has an explicit data label. That's one solution for the above problem.

Nevertheless, why not use another chart form that avoids these problems altogether?

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In Tyson's tweet, she showed another chart that pretty much contains the same information, this one from TASS.

This chart uses the flow diagram concept - in an abstract way, as I explained in previous post.

This chart form imposes structure on the data. The relative ranks of the countries within each region are listed from top to bottom. The relative amounts of grains are shown in black columns (and also in the thickness of the flows).

The aggregate value of movements within each region is called out in that middle section. It is impossible to learn this from the bag of bubbles version.

The designer did print the entire dataset onto this chart (except for the smallest countries grouped together as "other"). This decision takes away from the power of the underlying flow chart. Instead of thinking about the proportional representation of each country within its respective region, or the distribution of grains among regions, our eyes hone in on the data labels.

This brings me back to the principle of self-sufficiency: if we expect readers to consume the data labels - which comprise the entire dataset, why not just print a data table? If we decide to visualize, make the visual elements count!

Deconstructing graphics as an analysis tool in dataviz

One of the useful exercises I like to do with charts is to "deconstruct" them. (This amounts to a deeper version of the self-sufficiency test.)

Here is a chart stripped down to just the main visual elements.

The game is to guess what is the structure of the data given these visual elements.

I guessed the following:

• The data has a top-level split into two groups
• Within each group, the data is further split into 3 parts, corresponding to the 3 columns
• With each part, there are a variable number of subparts, each of which is given a unique color
• The color legend suggests that each group's data are split into 7 subparts, so I'm guessing that the 7 subparts are aggregated into 3 parts
• The core chart form is a stacked column chart with absolute values so relative proportions within each column (part) is important
• Comparing across columns is not supported because each column has its own total value
• Comparing same-color blocks across the two groups is meaningful. It's easier to compare their absolute values but harder to compare the relative values (proportions of total)

If I knew that the two groups are time periods, I'd also guess that the group on the left is the earlier time period, and the one on the right is the later time period. In addition to the usual left-to-right convention for time series, the columns are getting taller going left to right. Many things (not all, obviously) grow over time.

The color choice is a bit confusing because if the subparts are what I think they are, then it makes more sense to use one color and different shades within each column.

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The above guesses are a mixed bag. What one learns from the exercise is what cues readers are receiving from the visual structure.

Here is the same chart with key contextual information added back:

Now I see that the chart concerns revenues of a business over two years.

My guess on the direction of time was wrong. The more recent year is placed on the left, counter to convention. This entity therefore suffered a loss of revenues from 2017-8 to 2018-9.

The entity receives substantial government funding. In 2017-8, it has 1 dollar of government funds for every 2 dollars of revenues. In 2018-9, it's roughly 2 dollars of government funds per every 3 dollars of revenues. Thus, the ratio of government funding to revenues has increased.

On closer inspection, the 7 colors do not represent 7 components of this entity's funding. The categories listed in the color legend overlap.

It's rather confusing but I missed one very important feature of the chart in my first assessment: the three columns within each year group are nested. The second column breaks down revenues into 3 parts while the third column subdivides advertising revenues into two parts.

What we've found is that this design does not offer any visual cues to help readers understand how the three columns within a year-group relates to each other. Adding guiding lines or changing the color scheme helps.

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Next, I add back the data labels:

The system of labeling can be described as: label everything that is not further broken down into parts on the chart.

Because of the nested structure, this means two of the column segments, which are the sums of subparts, are not labeled. This creates a very strange appearance: usually, the largest parts are split into subparts, so such a labeling system means the largest parts/subparts are not labeled while the smaller, less influential, subparts are labeled!

You may notice another oddity. The pink segment is well above $1 billion but it is roughly the size of the third column, which represents $250 million. Thus, these columns are not drawn to scale. What happened? Keep reading.

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Here is the whole chart:

A twitter follower sent me this chart. Elon Musk has been feuding with the Canadian broadcaster CBC.

Notice the scale of the vertical axis. It has a discontinuity between $700 million and $1.7 billion. In other words, the two pink sections are artificially shortened. The erased section contains $1 billion (!) Notice that the erased section is larger than the visible section.

The focus of Musk's feud with CBC is on what proportion of the company's funds come from the government. On this chart, the only way to figure that out is to copy out the data and divide. It's roughly 1.2/1.7 = 70% approx.

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The exercise of deconstructing graphics helps us understand what parts are doing what, and it also reveals what cues certain parts send to readers.

In better dataviz, every part of the chart is doing something useful, it's free of redundant parts that take up processing time for no reason, and the cues to readers move them towards the intended message, not away from it.

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A couple of additional comments:

I'm not sure why old data was cited because in the most recent accounting report, the proportion of government funding was around 65%.

Source of funding is not a useful measure of pro- or anti-government bias, especially in a democracy where different parties lead the government at different times. There are plenty of mouthpiece media that do not apparently receive government funding.