Don't show everything

There are many examples where one should not show everything when visualizing data.

A long-time reader sent me this chart from the Economist, published around Thanksgiving last year:

It's a scatter plot with each dot representing a single tweet by Elon Musk against a grid of years (on the horizontal axis) and time of day (on the vertical axis).

The easy messages to pick up include:

• the increase in frequency of tweets over the years
• especially, the jump in density after Musk bought Twitter in late 2022 (there is also a less obvious level up around 2018)
• the almost continuous tweeting throughout 24 hours.

By contrast, it's hard if not impossible to learn the following:

• how many tweets did he make on average or in total per year, per day, per hour?
• the density of tweets for any single period of time (i.e., a reference for everything else)
• the growth rate over time, especially the magnitude of the jumps

The paradox: a chart that is data-dense but information-poor.

***

The designer added gridlines and axis labels to help structure our reading. Specifically, we're cued to separate the 24 hours into four 6-hour chunks. We're also expected to divide the years into two groups (pre- and post- the Musk acquisition), and secondarily, into one-year intervals.

If we accept this analytical frame, then we can divide time into these boxes, and then compute summary statistics within each box, and present those values.  I'm working on some concepts, will show them next time.

 

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.

***

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.

***

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.

***

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.

 

Stranger things found on scatter plots

Washington Post published a nice scatter plot which deconstructs scores from the recent World Championships in Gymnastics. (link)

The chart presents the main message clearly - the winner Simone Biles scored the highest on both components of the score (difficulty and execution), by quite some margin.

What else can we learn from this chart?

***

Every athlete who qualified for the final scored at or above average on both components.

Scoring below average on either component is a death knell: no athlete scored enough on the other component to compensate. (The top left and bottom right quadrants would have had some yellow dots otherwise.)

Several athletes in the top right quadrant presumably scored enough to qualify but didn't. The footnote likely explains it: each country can send at most two athletes to the final. It may be useful to mark out these "unlucky" athletes using a third color.

Curiously, it's not easy to figure out who these unlucky athletes were from this chart alone. We need two pieces of data: the minimum qualifying score, and the total score for each athlete. The scatter plot isn't the best chart form to show totals, but qualification to the final is based on the sum of the difficulty and execution scores. (Note also, neither axis starts at zero, compounding the challenge.)

***

This scatter plot is most memorable for shattering one of my expectations about risk and reward in sports.

I expect risk-seeking athletes to suffer from higher variance in performance. The tennis player who goes for big serves tend to also commit more double faults. The sluggers who hit home runs tend to strike out more often. Similarly, I expect gymnasts who attempt more difficult skills to receive lower execution scores.

Indeed, the headline writer seemed to agree, suggesting that Biles is special because she's both high in difficulty and strong in execution.

The scatter plot, however, sends the opposite message - this should not surprise. The entire field shows a curiously strong positive correlation between difficulty and execution scores. The more difficult is the routine, the higher the excution score!

It's hard to explain such a pattern. My guesses are:

a) judges reward difficult routines, and subconsciously confound execution and difficulty scores. They use separate judges for excecution and difficulty. Paradoxically, this arrangement may have caused separation anxiety - the judges for execution might just feel the urge to reward high difficulty.

b) those athletes who are skilled enough to attempt more difficult routines are also those who are more consistent in execution. This is a type of self-selection bias frequently found in observational data.

Regardless of the reasons for the strong correlation, the chart shows that these two components of the total score are not independent, i.e. the metrics have significant overlap in what they measure. Thus, one cannot really talk about a difficult routine without also noting that it's a well-executed routine, and vice versa. In an ideal scoring design, we'd like to have independent components.

The blue mist

The New York Times printed several charts about Twitter "blue checks," and they aren't one of their best efforts (link).

Blue checks used to be credentials given to legitimate accounts, typically associated with media outlets, celebrities, brands, professors, etc. They are free but must be approved by Twitter. Since Elon Musk acquired Twitter, he turned blue checks into a revenue generator. Yet another subscription service (but you're buying "freedom"!). Anyone can get a blue check for US$8 per month.

[The charts shown here are scanned from the printed edition.]

The first chart is a scatter plot showing the day of joining Twitter and the total number of followers the account has as of early November, 2022. Those are very strange things to pair up on a scatter plot but I get it: the designer could only work with the data that can be pulled down from Twitter's API.

What's wrong with the data? It would seem the interesting question is whether blue checks are associated with number of followers. The chart shows only Twitter Blue users so there is nothing to compare to. The day of joining Twitter is not the day of becoming "Twitter Blue", almost surely not for any user (Nevetheless, the former is not a standard data element released by Twitter). The chart has a built-in time bias since the longer an account exists, one would assume the higher the number of followers (assuming all else equal). Some kind of follower rate (e.g. number of followers per year of existence) might be more informative.

Still, it's hard to know what the chart is saying. That most Blue accounts have fewer than 5,000 followers? I also suspect that they chopped off the top of the chart (outliers) and forgot to mention it. Surely, some of the celebrity accounts have way over 150,000 followers. Another sign that the top of the chart was removed is that an expected funnel effect is not seen. Given the follower count is cumulative from the day of registration, we'd expect the accounts that started in the last few months should have markedly lower counts than those created years ago. (This is even more true if there is a survivorship bias - less successful accounts are more likely to be deleted over time.)

The designer arbitrarily labelled six specific accounts ("Crypto influencer", "HBO fan", etc.) but this feature risks sending readers the wrong message. There might be one HBO fan account that quickly grew to 150,000 followers in just a few months but does the data label suggest to readers that HBO fan accounts as a group tend to quickly attain high number of followers?

***

The second chart, which is an inset of the first, attempts to quantify the effect of the Musk acquisition on the number of "registrations and subscriptions". In the first chart, the story was described as "Elon Musk buys Twitter sparking waves of new users who later sign up for Twitter Blue".

The second chart confuses me. I was trying to figure out what is counted in the vertical axis. This was before I noticed the inset in the first chart, easy to miss as it is tucked into the lower right corner. I had presumed that the axis would be the same as in the first chart since there weren't any specific labels. In that case, I am looking at accounts with 0 to 500 followers, pretty inconsequential accounts. Then, the chart title uses the words "registrations and subscriptions." If the blue dots on this chart also refer to blue-check accounts as in the first chart, then I fail to see how this chart conveys any information about registrations (wbich presumably would include free accounts). As before, new accounts that aren't blue checks won't appear.

Further, to the extent that this chart shows a surge in subscriptions, we are restricted to accounts with fewer than 500 followers, and it's really unclear what proportion of total subscribers is depicted. Nor is it possible to estimate the magnitude of this surge.

Besides, I'm seeing similar densities of the dots across the entire time window between October 2021 and 2022. Perhaps the entire surge is hidden behind the black lines indicating the specific days when Musk announced and completed the acquisition, respectively. If the surge is hiding behind the black vertical lines, then this design manages to block the precise spots readers are supposed to notice.

Here is where we can use the self-sufficiency test. Imagine the same chart without the text. What story would you have learned from the graphical elements themselves? Not much, in my view.

***

The third chart isn't more insightful. This chart purportedly shows suspended accounts, only among blue-check accounts.

From what I could gather (and what I know about Twitter's API), the chart shows any Twitter Blue account that got suspended at any time. For example, all the black open circles occurring prior to October 27, 2022 represent suspensions by the previous management, and presumably have nothing to do with Elon Musk, or his decision to turn blue checks into a subscription product.

There appears to be a cluster of suspensions since Musk took over. I am not sure what that means. Certainly, it says he's not about "total freedom". Most of these suspended accounts have fewer than 50 followers, and only been around for a few weeks. And as before, I'm not sure why the analyst decided to focus on accounts with fewer than 500 followers.

What could have been? Given the number of suspended accounts are relatively small, an interesting analysis would be to form clusters of suspended accounts, and report on the change in what types of accounts got suspended before and after the change of management.

***

The online article (link) is longer, filling in some details missing from the printed edition.

There is one view that shows the larger accounts:

While more complete, this view isn't very helpful as the biggest accounts are located in the sparsest area of the chart. The data labels again pick out strange accounts like those of adult film stars and an Arabic news site. It's not clear if the designer is trying to tell us that most of Twitter Blue accounts belong to those categories.

***
See here for commentary on other New York Times graphics.

 

 

 

 

A graphical compass

A Twitter user pointed me to this article from Washington Post, ruminating about the correlation between gas prices and measures of political sentiment (such as Biden's approval rating or right-track-wrong-track). As common in this genre, the analyst proclaims that he has found something "counter intuitive".

The declarative statement strikes me as odd. In the first two paragraphs, he said the data showed "as gas prices fell, American optimism rose. As prices rose, optimism fell... This seems counterintuitive."

I'm struggling to see what's counterintuitive. Aren't the data suggesting people like lower prices? Is that not what we think people like?

The centerpiece of the article concerns the correlation between metrics. "If two numbers move in concert, they can be depicted literally moving in concert. One goes up, the other moves either up or down consistently." That's a confused statement and he qualifies it by typing "That sort of thing."

He's reacting to the following scatter plot with lines. The Twitter user presumably found it hard to understand. Count me in.

Why is this chart difficult to grasp?

The biggest puzzle is: what differentiates those two lines? The red and the gray lines are not labelled. One would have to consult the article to learn that the gray line represents the "raw" data at weekly intervals. The red line is aggregated data at monthly intervals. In other words, each red dot is an average of 4 or 5 weekly data points. The red line is just a smoothed version of the gray line. Smoothed lines show the time trend better.

The next missing piece is the direction of time, which can only be inferred by reading the month labels on the red line. But the chart without the direction of time is like a map without a compass. Take this segment for example:

If time is running up to down, then approval ratings are increasing over time while gas prices are decreasing. If time is running down to up, then approval ratings are decreasing over time while gas prices are increasing. Exactly the opposite!

The labels on the red line are not sufficient. It's possible that time runs in the opposite direction on the gray line! We only exclude that possibility if we know that the red line is a smoothed version of the gray line.

This type of chart benefits from having a compass. Here's one:

It's useful for readers to know that the southeast direction is "good" (higher approval ratings, lower gas prices) while the northwest direction is "bad". Going back to the original chart, one can see that the metrics went in the "bad" direction at the start of the year and has reverted to a "good" direction since.

***

What does this chart really say? The author remarked that "correlation is not causation". "Just because Biden’s approval rose as prices dropped doesn’t mean prices caused the drop."

Here's an alternative: People have general sentiments. When they feel good, they respond more positively to polls, as in they rate everything more positively. The approval ratings are at least partially driven by this general sentiment. The same author apparently has another article saying that the right-track-wrong-track sentiment also moved in tandem with gas prices.

One issue with this type of scatter plot is that it always cues readers to make an incorrect assumption: that the outcome variables (approval rating) is solely - or predominantly - driven by the one factor being visualized (gas prices). This visual choice completely biases the reader's perception.

P.S. [11-11-22] The source of the submission was incorrectly attributed.

Painting the corner

Found an old one sitting in my folder. This came from the Wall Street Journal in 2018.

At first glance, the chart looks like a pretty decent effort.

The scatter plot shows Ebitda against market value, both measured in billions of dollars. The placement of the vertical axis title on the far side is a little unusual.

Ebitda is a measure of business profit (something for a different post on the sister blog: the "b" in Ebitda means "before", and allows management to paint a picture of profits without accounting for the entire cost of running the business). In the financial markets, the market value is claimed to represent a "fair" assessment of the value of the business. The ratio of the market value to Ebitda is known as the "Ebitda multiple", which describes the number of dollars the "market" places on each dollar of Ebitda profit earned by the company.

Almost all scatter plots suffer from xyopia: the chart form encourages readers to take an overly simplistic view in which the market cares about one and only one business metric (Ebitda). The reality is that the market value contains information about Ebitda plus lots of other factors, such as competitors, growth potential, etc.

Consider Alphabet vs AT&T. On this chart, both companies have about $50 billion in Ebitda profits. However, the market value of Alphabet (Google's mother company) is about four times higher than that of AT&T. This excess valuation has nothing to do with profitability but partly explained by the market's view that Google has greater growth potential.

***

Unusually, the desginer chose not to utilize the log scale. The right side of the following display is the same chart with a log horizontal axis.

The big market values are artificially pulled into the middle while the small values are plied apart. As one reads from left to right, the same amount of distance represents more and more dollars. While all data visualization books love log scales, I am not a big fan of it. That's because the human brain doesn't process spatial information this way. We don't tend to think in terms of continuously evolving scales. Thus, presenting the log view causes readers to underestimate large values and overestimate small differences.

Now let's get to the main interest of this chart. Notice the bar chart shown on the top right, which by itself is very strange. The colors of the bar chart is coordinated with those on the scatter plot, as the colors divide the companies into two groups; "media" companies (old, red), and tech companies (new, orange).

Scratch that. Netflix is found in the scatter plot but with a red color while AT&T and Verizon appear on the scatter plot as orange dots. So it appears that the colors mean different things on different plots. As far as I could tell, on the scatter plot, the orange dots are companies with over $30 billion in Ebitda profits.

At this point, you may have noticed the stray orange dot. Look carefully at the top right corner, above the bar chart, and you'll find the orange dot representing Apple. It is by far the most important datum, the company that has the greatest market value and the largest Ebitda.

I'm not sure burying Apple in the corner was a feature or a bug. It really makes little sense to insert the bar chart where it is, creating a gulf between Apple and the rest of the companies. This placement draws the most attention away from the datum that demands the most attention.

 

 

 

Another reminder that aggregate trends hide information

The last time I looked at the U.S. employment situation, it was during the pandemic. The data revealed the deep flaws of the so-called "not in labor force" classification. This classification is used to dehumanize unemployed people who are declared "not in labor force," in which case they are neither employed nor unemployed -- just not counted at all in the official unemployment (or employment) statistics.

The reason given for such a designation was that some people just have no interest in working, or even looking for a job. Now they are not merely discouraged - as there is a category of those people. In theory, these people haven't been looking for a job for so long that they are no longer visible to the bean counters at the Bureau of Labor Statistics.

What happened when the pandemic precipitated a shutdown in many major cities across America? The number of "not in labor force" shot up instantly, literally within a few weeks. That makes a mockery of the reason for such a designation. See this post for more.

***

The data we saw last time was up to April, 2020. That's more than two years old.

So I have updated the charts to show what has happened in the last couple of years.

Here is the overall picture.

In this new version, I centered the chart at the 1990 data. The chart features two key drivers of the headline unemployment rate - the proportion of people designated "invisible", and the proportion of those who are considered "employed" who are "part-time" workers.

The last two recessions have caused structural changes to the labor market. From 1990 to late 2000s, which included the dot-com bust, these two metrics circulated within a small area of the chart. The Great Recession of late 2000s led to a huge jump in the proportion called "invisible". It also pushed the proportion of part-timers to all0time highs. The proportion of part-timers has fallen although it is hard to interpret from this chart alone - because if the newly invisible were previously part-time employed, then the same cause can be responsible for either trend.

Readers of Numbersense (link) might be reminded of a trick used by school deans to pump up their US News rankings. Some schools accept lots of transfer students. This subpopulation is invisible to the US News statisticians since they do not factor into the rankings. The recent scandal at Columbia University also involves reclassifying students (see this post).

Zooming in on the last two years. It appears that the pandemic-related unemployment situation has reversed.

***

Let's split the data by gender.

American men have been stuck in a negative spiral since the 1990s. With each recession, a higher proportion of men are designated BLS invisibles.

In the grid system set up in this scatter plot, the top right corner is the worse of all worlds - the work force has shrunken and there are more part-timers among those counted as employed. The U.S. men are not exiting this quadrant any time soon.

***
What about the women?

If we compare 1990 with 2022, the story is not bad. The female work force is gradually reaching the same scale as in 1990 while the proportion of part-time workers have declined.

However, celebrating the above is to ignore the tremendous gains American women made in the 1990s and 2000s. In 1990, only 58% of women are considered part of the work force - the other 42% are not working but they are not counted as unemployed. By 2000, the female work force has expanded to include about 60% with similar proportions counted as part-time employed as in 1990. That's great news.

The Great Recession of the late 2000s changed that picture. Just like men, many women became invisible to BLS. The invisible proportion reached 44% in 2015 and have not returned to anywhere near the 2000 level. Fewer women are counted as part-time employed; as I said above, it's hard to tell whether this is because the women exiting the work force previously worked part-time.

***

The color of the dots in all charts are determined by the headline unemployment number. Blue represents low unemployment. During the 1990-2022 period, there are three moments in which unemployment is reported as 4 percent or lower. These charts are intended to show that an aggregate statistic hides a lot of information. The three times at which unemployment rate reached historic lows represent three very different situations, if one were to consider the sizes of the work force and the number of part-time workers.

 

P.S. [8-15-2022] Some more background about the visualization can be found in prior posts on the blog: here is the introduction, and here's one that breaks it down by race. Chapter 6 of Numbersense (link) gets into the details of how unemployment rate is computed, and the implications of the choices BLS made.

P.S. [8-16-2022] Corrected the axis title on the charts (see comment below). Also, added source of data label.

Funnels and scatters

I took a peek at some of the work submitted by Ray Vella's students in his NYU dataviz class recently.

The following chart by Hosanah Bryan caught my eye:

The data concern the GDP gap between rich and poor regions in various countries. In some countries, especially in the U.K., the gap is gigantic. In other countries, like Spain and Sweden, the gap is much smaller.

The above chart uses a funnel metaphor to organize the data, although the funnel does not add more meaning (not that it has to). Between that, the color scheme and the placement of text, it's visually clean and pleasant to look at.

The data being plotted are messy. They are not actual currency values of GDP. Each number is an index, and represents the relative level of the GDP gap in a given year and country. The gap being shown by the colored bars are differences in these indices 15 years apart. (The students were given this dataset to work with.)

So the chart is very hard to understand if one focuses on the underlying data. Nevertheless, the same visual form can hold other datasets which are less complicated.

One can nitpick about the slight misrepresentation of the values due to the slanted edges on both sides of the bars. This is yet another instance of the tradeoff between beauty and precision.

***

The next chart by Liz Delessert engages my mind for a different reason.

The scatter plot sets up four quadrants. The top right is "everyone gets richer". The top left, where most of the dots lie, is where "the rich get richer, the poor get poorer".  This chart shows a thoughtfulness about organizing the data, and the story-telling.

The grid setup cues readers toward a particular way of looking at the data.

But power comes with responsibility. Such scatter plots are particularly susceptible to the choice of data, in this case, countries. It is tempting to conclude that there are no countries in which everyone gets poorer. But that statement more likely tells us more about which countries were chosen than the real story.

I like to see the chart applied to other data transformations that are easier. For example, we can start with the % change in GDP computed separately for rich and for poor. Then we can form a ratio of these two percent changes.

 

 

What does Elon Musk do every day?

The Wall Street Journal published a fun little piece about tweets by Elon Musk (link).

Here is an overview of every tweet he sent since he started using Twitter more than a decade ago.

Apparently, he sent at least one tweet almost every day for the last four years. In addition, his tweets appear at all hours of the day. (Presumably, he is not the only one tweeting from his account.)

He doesn't just spend time writing tweets; he also reads other people's tweets. WSJ finds that up to 80% of his tweets include mentions of other users.

***

One problem with "big data" analytics is that they often don't answer interesting questions. Twitter is already one of the companies that put more of their data out there, but still, analysts are missing some of the most important variables.

We know that Musk has 93 million followers. We already know from recent news that a large proportion of such users may be spam/fake. It is frequently assumed in twitter analysis that any tweet he makes reaches 93 million accounts. That's actually far from correct. Twitter uses algorithms to decide what posts show up in each user's feed so we have no idea how many of the 93 million accounts are in fact exposed to any of Musk's tweets.

Further, not every user reads everything on their Twitter feed. I don't even check it every day. Because Twitter operates as a 'firehose" with ever-changing content as users send out short messages at all hours, what one sees depends on when one reads. If Musk tweets in the morning, the users who log on in the afternoon won't see it.

Let's say an analyst wants to learn how impactful Musk's tweets are. That's pretty difficult when one can't figure out which of the 93 million followers were shown these tweets, and who read them. The typical data used to measure response are retweets and likes. Those are convenient metrics because they are available. They are very limited in what they measure. There are lots of users who don't like or retweet at all.

***

The available data do make for some fun charts. This one gave me a big smile:

Between writing tweets, reading tweets, and ROTFL, every hour of almost every day, Musk finds time to run his several companies. That's impressive.

 

The gift of small edits and subtraction

While making the chart on fertility rates (link), I came across a problem that pops up quite often, and is  ignored by most software programs.

Here is an earlier version of the chart I later discarded:

Compare this to the version I published in the blog post:

Aside from adding the chart title, there is one major change. I removed the empty plots from the grid. This is a visualization trick that should be called adding by subtracting. The empty scaffolding on the first chart increases our cognitive load without yielding any benefit. The whitespace brings out the message that only countries in Asia and Africa have fertility rates above 5.0. 

This is a small edit. But small edits accumulate and deliver a big impact. Bear this in mind the next time you make a chart.

 

P.S.

(1) You'd have to use a lower-level coding language to execute this small edit. Most software programs are quite rigid when it comes to making small-multiples (facet) charts.

(2) If there is a next iteration, I'd reverse the Asia and Oceania rows.