Color bomb

I found a snapshot of the following leaderboard (link) in a newsletter in my inbox.

This chart ranks different AIs (foundational models) by token usage (which is the unit by which AI companies charge users).

It's a standard stacked column chart, with data aggregated by week. The colors represent different foundational models.

In the original webpage, there is a table printed below, listing the top 20 model names, ordered from the most tokens used.

Certain AI models have come and gone (e.g. the yellow and blue ones at the bottom of the chart in the first half). The model in pink has been the front runner through all weeks.

Total usage has been rising, although it might be flattening, which is the point made by the newsletter publisher.

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A curiosity is the gray shaded section on the far right - it represents the projected total token usage for the days that have not yet passed during the current week. This is one of those additions that I like to see more often. If the developer had chosen to plot the raw data and nothing more, then they would have made the same chart except for the gray section. On that chart, the last column should not be compared to any other column as it is the only one that encodes a partial week.

This added gray section addresses the specific question: whether the total token usage for the current week is on pace with prior weeks, or faster or slower. (The accuracy of the projection is a different matter, which I won't discuss.)

This added gray section leaves another set of questions unanswered. The chart suggests that the total token usage is expected to exceed the values for the prior few weeks, at the time it was frozen. We naturally want to know which models are contributing to this projected growth (and which aren't). The current design cannot address this issue because the projected additional usage is aggregated, and not available at the model level.

While it "tops up" the weekly total usage using a projected value, the chart does not show how many days are remaining. That's an important piece of information for interpreting the projection.

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Now, we come to the good part, for those of us who loves details.

A major weakness of these stacked column charts is of course the dizzy set of colors required, one for each model. Some of the shades are so similar it's hard to tell if they repeated colors. Are these two different blues or the same blue?

Besides, the visualization software has a built-in feature that "softens" a color when it is clicked on. This feature introduces unpleasant surprises as that soft shade might have been used for another category.

It appears that the series is running sideways (following the superimposed gray line) when in fact the first section is a softened red associated with the series that went higher (following the white line).

It's near impossible to work with so many colors. If you extract the underlying data, you find that they show 10 values per day across 24 weeks. Because the AI companies are busy launching new models, the dataset contains 40 unique model names, which imply they needed 40 different shades on this one chart. (Double that to 80 shades if we add the colors on click variations.)

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I hope some of you have noticed something else. Earlier, I mentioned the model in pink as the most popular AI model but if you take a closer look, this pink section actually represents a mostly useless catch-all category called "Others," that presumably aggregates the token usages of a range of less popular models. In this design, the Others category is catching an undeserved amount of attention.

It's unclear how the models are ordered within each column. The developer did not group together different generations of models by the same developer. Anthropic Claude has many entries: Sonnet 4 [green], Sonnet 3.5 [blue], Sonnet 3.5 (self-moderated) [yellow], Sonnet 3.7 (thinking) [pink], Sonnet 3.7 [violet], Sonnet 3.7 (self-moderated) [cyan], etc. The same for OpenAI, Google, etc.

This graphical decision may reflect how users of large language models evaluate performance. Perhaps at this time, there is no brand loyalty, or lock-in effect, and users see all these different models as direct substitutes. Therefore, our attention is focused on the larger number of individual models, rather than the smaller set of AI developers.

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Before ending the post, I must point out that the publisher of this set of rankings offers a platform that allows users to switch between models. They are visualizing their internal data. This means the dataset only describes what customers of Openrouter.ai do on this platform. There should be no expectation that this company's user base is representative of all users of LLMs.

Students demonstrate how analytics underlie strong dataviz

In today's post, I'm delighted to feature work by several students of Ray Vella's data visualization class at NYU. They have been asked to improve the following Economist chart entitled "The Rich Get Richer".

In my guest lecture to the class, I emphasized the importance of upfront analytics when constructing data visualizations.

One of the key messages is pay attention to definitions. How does the Economist define "rich" and "poor"? (it's not what you think). Instead of using percentiles (e.g. top 1% of the income distribution), they define "rich" as people living in the richest region by average GDP, and "poor" as people living in the poorest region by average GDP. Thus, the "gap" between the rich and the poor is measured by the difference in GDP between the average persons in those two regions.

I don't like this metric at all but we'll just have to accept that that's the data available for the class assignment.

***

Shulin Huang's work is notable in how she clarifies the underlying algebra.

The middle section classifies the countries into two groups, those with widening vs narrowing gaps. The side panels show the two components of the gap change. The gap change is the sum of the change in the richest region and the change in the poorest region.

If we take the U.S. as an example, the gap increased by 1976 units. This is because the richest region gained 1777 while the poor region lost 199. Germany has a very different experience: the richest region regressed by 2215 while the poorest region improved by 424, leading to the gap narrowing by 2638.

Note how important it is to keep the order of the countries fixed across all three panels. I'm not sure how she decided the order of these countries, which is a small oversight in an otherwise excellent effort.

Shulin's text is very thoughtful throughout. The chart title clearly states "rich regions" rather than "the rich". Take a look at the bottom of the side panels. The label "national AVG" shows that the zero level is the national average. Then, the label "regions pulled further ahead" perfectly captures the positive direction.

Compared to the original, this chart is much more easily understood. The secret is the clarity of thought, the deep understanding of the nature of the data.

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Michael Unger focuses his work on elucidating the indexing strategy employed by the Economist. In the original, each value of regional average GDP is indexed to the national average of the relevant year. A number like 150 means the region has an average GDP for the given year that is 50% higher than the national average. It's tough to explain how such indices work.

Michael's revision goes back to the raw data. He presents them in two panels. On the left, the absolute change over time in the average GDPs are presented for each of the richest/poorest region while on the right, the relative change is shown.

(Some of the country labels are incorrect. I'll replace with a corrected version when I receive one.)

Presenting both sides is not redundant. In France, for example, the richest region improved by 17K while the poorest region went up by not quite 6K. But 6K on a much lower base represents a much higher proportional jump as the right side shows.

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Related to Michael's work, but even simpler, is Debbie Hsieh's effort.

Debbie reduces the entire exercise to one message - the relative change over time in average GDP between the richest and poorest region in each country. In this simplest presentation, if both columns point up, then both the richest and the poorest region increased their average GDP; if both point down, then both regions suffered GDP drops.

If the GDP increased in the richest region while it decreased in the poorest region, then the gap widened by the most. This is represented by the blue column pointing up and the red column pointing down.

In some countries (e.g. Sweden), the poorest region (orange) got worse while the richest region (blue) improved slightly. In Italy and Spain, both the best and worst regions gained in average GDPs although the richest region attained a greater relative gain.

While Debbie's chart is simpler, it hides something that Michael's work shows more clearly. If both the richest and poorest regions increased GDP by the same percentage amount, the average person in the richest region actually experienced a higher absolute increase because the base of the percentage is higher.

***

The numbers across these charts aren't necessarily well aligned. That's actually one of the challenges of this dataset. There are many ways to process the data, and small differences in how each student handles the data lead to differences in the derived values, resulting in differences in the visual effects.

Aligning the visual and the message

Today's post is about work by Diane Barnhart, who is a product manager at Bloomberg, and is taking Ray Vella's infographics class at NYU. The class is given a chart from the Economist, as well as some data on GDP per capita in selected countries at the regional level. The students are asked to produce data visualization that explores the change in income inequality (as indicated by GDP per capita).

Here is Diane's work:

In this chart, the key measure is the GDP per capita of different regions in Germany relative to the national average GDP. Hamburg, for example, has a GDP per capita that was 80% above the national average in 2000 while Leipzig's GDP per capita was 30% below the national average in 2000. (This metric is a bit of a head scratcher, and forms the basis of the Economist chart.)

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Diane made several insightful design choices.

The key insight of this graph is also one of the easiest to see. It's the narrowing of the range of possible values. In 2000, the top value is about 90% while the bottom is under -40%, making a range of 130%. In 2020, the range has narrowed to 90%, with the values falling between 60% and -30%. In other words, the gap between rich and poor regions in Germany has reduced over these two decades.

The chosen chart form makes this message come alive.

Diane divided the regions into three groups, mapped to the black, red and yellow colors of the German flag. Black are for those regions that have GDP per capita above the average; yellow for those regions with GDP per capita over 25% below the average.

Instead of applying color to individual lines that trace the GDP metric over time for each region, she divided the area between the lines into three, and painted them. This necessitates a definition of the boundary line between colored areas over time. I gathered that she classified the regions using the latest GDP data (2020) and then traced the GDP trend lines back in time. Other definitions are also possible.

The two-column data table shown on the right provides further details that aren't found in the data visualization. The table is nicely enhanced with colors. They represent an augmentation of the information in the main chart, not a repetition.

All in all, this is a delightful project, and worthy of a top grade!

Organizing time-stamped data

In a previous post, I looked at the Economist chart about Elon Musk's tweeting compulsion. It's chart that contains lots of data, every tweet included, but one can't tell the number or frequency of tweets.

In today's post, I'll walk through a couple of sketches of other charts. I was able to find a dataset on Github that does not cover the same period of time but it's good enough for illustration purposes.

As discussed previously, I took cues from the Economist chart, in particular that the hours of the day should be divided up into four equal-width periods. One thing Musk is known for is tweeting at any hour of the day.

This is a small-multiples arrangement of column charts. Each column chart represents the tweets that were posted during a six-hour window, across all days in the dataset. A column covers half a year of tweets. We note that there were more tweets in the afternoon hours as he started tweeting more. In the first half of 2022, he sent roughly 750 tweets between 7 pm and midnight.

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In this next sketch, I used a small-multiples of line charts. Each line chart represents tweets posted during a six-hour window, as before. Instead of counting how many tweets, here I "smoothed" the daily tweet count, so that each number is an average daily tweet count, with the average computed based on a rolling time window.

 

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Finally, let's cover a few details only people who make charts would care about. The time of day variable only makes sense if all times are expressed as "local time", i.e. the time at the location where Musk was tweeting from. This knowledge is not necessary to make a chart but it is essential to make the chart interpretable. A statement like Musk tweets a lot around midnight assumes that it was midnight where he was when he sent each tweet.

Since we don't have his travel schedule, we will definitely be wrong. In my charts, I assumed he is in the Pacific time zone, and never tweeted anywhere outside that time zone.

(Food for thought: the server that posts tweets certainly had the record of the time and time zone for each tweet. Typically, databases store these time stamps standardized to one time zone - call it Greenwich Mean Time. If you have all time stamps expressed in GMT, is it now possible to make a statement about midnight tweeting? Does standardizing to one time zone solve this problem?)

In addition, I suspect that there may be problems with the function used to compute those rolling sums and averages, so take the actual numbers on those sketches with a grain of salt. Specifically, it's hard to tell on any of these charts but Musk did not tweet every single day so there are lots of holes in the time series.

Ranks, labels, metrics, data and alignment

A long-time reader Chris V. (since 2012!) sent me to this WSJ article on airline ratings (link).

The key chart form is this:

It's a rhombus shaped chart, really a bar chart rotated counter-clockwise by 45 degrees. Thus, all the text is at 45 degree angles. An airplane icon is imprinted on each bar.

There is also this cute interpretation of the white (non-data-ink) space as a symmetric reflection of the bars (with one missing element). On second thought, the decision to tilt the chart was probably made in service of this quasi-symmetry. If the data bars were horizontal, then the white space would have been sliced up into columns, which just doesn't hold the same appeal.

If we be Tuftian, all of these flourishes do not serve the data. But do they do much harm? This is a case that's harder to decide. The data consist of just a ranking of airlines. The message still comes across. The head must tilt, but the chart beguiles.

***

As the article progresses, the same chart form shows up again and again, with added layers of detail. I appreciate how the author has constructed the story. Subtly, the first chart teaches the readers how the graphic encodes the data, and fills in contextual information such as there being nine airlines in the ranking table.

In the second section, the same chart form is used, while the usage has evolved. There are now a pair of these rhombuses. Each rhombus shows the rankings of a single airline while each bar inside the rhombus shows the airline's ranking on a specific metric. Contrast this with the first chart, where each bar is an airline, and the ranking is the overall ranking on all metrics.

You may notice that you've used a piece of knowledge picked up from the first chart - that on each of these metrics, each airline has been ranked against eight others. Without that knowledge, we don't know that being 4th is just better than the median. So, in a sense, this second section is dependent on the first chart.

There is a nice use of layering, which links up both charts. A dividing line is drawn between the first place (blue) and not being first (gray). This layering allows us to quickly see that Delta, the overall winner, came first in two of the seven metrics while Southwest, the second-place airline, came first in three of the seven (leaving two metrics for which neither of these airlines came first).

I'd be the first to admit that I have motion sickness. I wonder how many of you are starting to feel dizzy while you read the labels, heads tilted. Maybe you're trying, like me, to figure out the asterisks and daggers.

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Ironically, but not surprisingly, the asterisks reveal a non-trivial matter. Asterisks direct readers to footnotes, which should be supplementary text that adds color to the main text without altering its core meaning. Nowadays, asterisks may hide information that changes how one interprets the main text, such as complications that muddy the main argument.

Here, the asterisks address a shortcoming of representing ranking using bars. By convention, lower ranking indicates better, and most ranking schemes start counting from 1. If ranks are directly encoded in bars, then the best object is given the shortest bar. But that's not what we see on the chart. The bars actually encode the reverse ranking so the longest bar represents the lowest ranking.

That's level one of this complication. Level two is where these asterisks are at.

Notice that the second metric is called "Canceled flights". The asterisk quipped "fewest". The data collected is on the number of canceled flights but the performance metric for the ranking is really "fewest canceled flights". 

If we see a long bar labelled "1st" under "canceled flights", it causes a moment of pause. Is the airline ranked first because it had the most canceled flights? That would imply being first is worst for this category. It couldn't be that. So perhaps "1st" means having the fewest canceled flights but then it's just weird to show that using the longest bar. The designer correctly anticipates this moment of pause, and that's why the chart has those asterisks.

Unfortunately, six out of the seven metrics require asterisks. In almost every case, we have to think in reverse. "Extreme delays" really mean "Least extreme delays"; "Mishandled baggage" really mean "Less mishandled baggage"; etc. I'd spend some time renaming the metrics to try to fix this avoiding footnotes. For example, saying "Baggage handling" instead of "mishandled baggage" is sufficient.

***

The third section contains the greatest details. Now, each chart prints the ranking of nine airlines for a particular metric.

 

By now, the cuteness faded while the neck muscles paid. Those nice annotations, written horizontally, offered but a twee respite.

 

 

 

 

 

Dot plots with varying dot sizes

In a prior post, I appreciated the effort by the Bloomberg Graphics team to describe the diverging fortunes of Japanese and Chinese car manufacturers in various Asian markets.

The most complex chart used in that feature is the following variant of a dot plot:

This chart plots the competitors in the Chinese domestic car market. Each bubble represents a car brand. Using the styling of the entire article, the red color is associated with Japanese brands while the medium gray color indicates Chinese brands. The light gray color shows brands from the rest of the world. (In my view, adding the pink for U.S. and blue for German brands - seen on the first chart in this series - isn't too much.)

The dot size represents the current relative market share of the brand. The main concern of the Bloomberg article is the change in market share in the period 2019-2024. This is placed on the horizontal axis, so the bubbles on the right side represent growing brands while the bubbles on the left, weakening brands.

All the Japanese brands are stagnating or declining, from the perspective of market share.

The biggest loser appears to be Volkswagen although it evidently started off at a high level since its bubble size after shrinkage is still among the largest.

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This chart form is a composite. There are at least two ways to describe it. I prefer to see it as a dot plot with an added dimension of dot size. A dot plot typically plots a single dimension on a single axis, and here, a second dimension is encoded in the sizes of the dots.

An alternative interpretation is that it is a scatter plot with a third dimension in the dot size. Here, the vertical dimension is meaningless, as the dots are arbitrarily spread out to prevent overplotting. This arrangement is also called the bubble plot if we adopt a convention that a bubble is a dot of variable size. In a typical bubble plot, both vertical and horizontal axes carry meaning but here, the vertical axis is arbitrary.

The bubble plot draws attention to the variable in the bubble size, the scatter plot emphasizes two variables encoded in the grid while the dot plot highlights a single metric. Each shows secondary metrics.

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Another revelation of the graph is the fragmentation of the market. There are many dots, especially medium gray dots. There are quite a few Chinese local manufacturers, most of which experienced moderate growth. Most of these brands are startups - this can be inferred because the size of the dot is about the same as the change in market share.

The only foreign manufacturer to make material gains in the Chinese market is Tesla.

The real story of the chart is BYD. I almost missed its dot on first impression, as it sits on the far right edge of the chart (in the original webpage, the right edge of the chart is aligned with the right edge of the text). BYD is the fastest growing brand in China, and its top brand. The pedestrian gray color chosen for Chinese brands probably didn't help. Besides, I had a little trouble figuring out if the BYD bubble is larger than the largest bubble in the size legend shown on the opposite end of BYD. (I measured, and indeed the BYD bubble is slightly larger.)

This dot chart (with variable dot sizes) is nice for highlighting individual brands. But it doesn't show aggregates. One of the callouts on the chart reads: "Chinese cars' share rose by 23%, with BYD at the forefront". These words are necessary because it's impossible to figure out that the total share gain by all Chinese brands is 23% from this chart form.

They present this information in the line chart that I included in the last post, repeated here:

The first chart shows that cumulatively, Chinese brands have increased their share of the Chinese market by 23 percent while Japanese brands have ceded about 9 percent of market share.

The individual-brand view offers other insights that can't be found in the aggregate line chart. We can see that in addition to BYD, there are a few local brands that have similar market shares as Tesla.

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It's tough to find a single chart that brings out insights at several levels of analysis, which is why we like to talk about a "visual story" which typically comprises a sequence of charts.

 

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.

When should we use bar charts?

Two innocent looking column charts.

These came from an article in Significance magazine (link to paywall) that applies the "difference-in-difference" technique to analyze whether the superstitious act of skipping the number 13 when numbering floors in tall buildings causes an inflation of condo pricing.

The study authors are quite careful in their analysis, recognizing that building managers who decide to relabel the 13th floor as 14th may differ in other systematic ways from those who don't relabel. They use a matching technique to construct comparison groups. The left-side chart shows one effect of matching buildings, which narrowed the gap in average square footage between the relabeled and non-relabeled groups. (Any such gap suggests potential confounding; in a hypothetical, randomized experiment, the average square footage of both groups should be statistically identical.)

The left-side chart features columns that don't start as zero, thus the visualization exaggerates the differences. The degree of exaggeration here is tame: about 150 got chopped off at the bottom, which is about 10% of the total height. But why?

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The right-side chart is even more problematic.

This chart shows the effect of matching buildings on the average age of the buildings (measured using the average construction year). Again, the columns don't start at zero. But for this dataset, zero is a meaningless value. Never make a column chart when the zero level has no meaning!

The story is simple: by matching, the average construction year in the relabeled group was brought closer to that in the non-relabeled group. The construction year is an ordinal categorical variable, with integer values. I think a comparison of two histograms will show the message clearer, and also provide more information than jut the two average values.

Is this dataviz?

The message in this Visual Capitalist chart is simple - that big tech firms are spending a lot of cash buying back their own stock (which reduces the number of shares in the market, which pushes up their stock price - all without actually having improved their business results.)

But is this data visualization? How does the visual design reflect the data?

The chart form is a half-pie chart, composed of five sectors, of increasing radii. In a pie chart, the data are encoded in the sector areas. But when the sectors are of different radii, it's possible that the data are found in the angles.

The text along the perimeter, coupled with the bracketing, suggests that the angles convey information - specifically, the amount of shares repurchased as a proportion of outstanding share value (market cap). On inspection, the angles are the same for all five sectors, and each one is 180 degrees divided by five, the number of companies depicted on the chart, so they convey no information, unless the company tally is deemed informative.

Each slice of the pie represents a proportion but these proportions don't add up. So the chart isn't even a half-pie chart. (Speaking of which, should the proportions in a half-pie add up to 100% or 50%?)

What about the sector areas? Since the angles are fixed, the sector areas are directly proportional to the radii. It took me a bit of time to figure this one out. The radius actually encodes the amount spent by each company on the buyback transaction. Take the ratio of Microsoft to Meta: 20 over 25 is 80%. To obtain a ratio of areas of 80%, the ratio of radii is roughly 90%; and the radius of Microsoft's sector is indeed about 90% of that of Meta. The ratio between Alphabet and Apple is similar.

The sector areas represent the dollar value of these share buybacks, although these transactions range from 0.6% to 2.9% as a proportion of outstanding share value.

Here is a more straightforward presentation of the data:

I'm not suggesting using this display. The sector areas in the original chart depict the data in the red bars. It's not clear to me how the story is affected by the inclusion of the market value data (gray bars).

Reading log: HBR's specialty bar charts

Today, I want to talk about a type of analysis that I used to ask students to do. I'm calling it a reading log analysis – it's a reading report that traces how one consumes a dataviz work from where your eyes first land to the moment of full comprehension (or abandonment, if that is the outcome). Usually, we do this orally during a live session, but it's difficult to arrive at a full report within the limited class time. A written report overcomes this problem. A stack of reading logs should be a gift to any chart designer.

My report below is very detailed, reflecting the amount of attention I pay to the craft. Most readers won't spend as much time consuming a graphic. The value of the report is not only in what it covers but also in what it does not mention.

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The chart being analyzed showed up in a Harvard Business Review article (link), and it was submitted by longtime reader Howie H.

First and foremost, I recognized the chart form as a bar chart. It's an advanced bar chart in which each bar has stacked sections and a vertical line in the middle. Now, I wanted to figure out how data enter the picture.

My eyes went to the top legend which tells me the author was comparing the proportion of respondents who said "business should take responsibility" to the proportion who rated "business is doing well". The difference in proportions is called the "performance gap". I glanced quickly at the first row label to discover the underlying survey addresses social issues such as environmental concerns.

Next, I looked at the first bar, trying to figure out its data encoding scheme. The bold, blue vertical line in the middle of the bar caused me to think each bar is split into left and right sections. The right section is shaded and labeled with the performance gap numbers so I focused on the segment to the left of the blue line.

My head started to hurt a little. The green number (76%) is associated with the left edge of the left section of the bar. And if the blue line represents the other number (29%), then the width of the left section should map to the performance gap. This interpretation was obviously incorrect since the right section already showed the gap, and the width of the left section was not equal to that of the right shaded section.

I jumped to the next row. My head hurt a little bit more. The only difference between the two rows is the green number being 74%, 2 percent smaller. I couldn't explain how the left sections of both bars have the same width, which confirms that the left section doesn't display the performance gap (assuming that no graphical mistakes have been made). It also appeared that the left edge of the bar was unrelated to the green number. So I retreated to square one. Let's start over. How were the data encoded in this bar chart?

I scrolled down to the next figure, which applies the same chart form to other data.

I became even more confused. The first row showed labels (green number 60%, blue number 44%, performance gap -16%). This bar is much bigger than the one in the previous figure, even though 60% was less than 76%. Besides, the left section, which is bracketed by the green number on the left and the blue number on the right, appeared much wider than the 16% difference that would have been merited. I again lapsed into thinking that the left section represents performance gaps.

Then I noticed that the vertical blue lines were roughly in proportion. Soon, I realized that the total bar width (both sections) maps to the green number. Now back to the first figure. The proportion of respondents who believe business should take responsibility (green number) is encoded in the full bar. In other words, the left edges of all the bars represent 0%. Meanwhile the proportion saying business is doing well is encoded in the left section. Thus, the difference between the full width and the left-section width is both the right-section width and the performance gap.

Here is an edited version that clarifies the encoding scheme:

***

That's my reading log. Howie gave me his take:

I had to interrupt my reading of the article for quite a while to puzzle this one out. It's sorted by performance gap, and I'm sure there's a better way to display that. Maybe a dot plot, similar to here - https://junkcharts.typepad.com/junk_charts/2023/12/the-efficiency-of-visual-communications.html.

A dot plot might look something like this:

Howie also said:

I interpret the authros' gist to be something like "Companies underperform public expectations on a wide range of social challenges" so I think I'd want to focus on the uniform direction and breadth of the performance gap more than the specifics of each line item.

And I agree.