## The radial is still broken

##### Jun 21, 2024

It's puzzling to me why people like radial charts. Here is a recent set of radial charts that appear in an article in Significance magazine (link to paywall, currently), analyzing NBA basketball data.

This example is not as bad as usual (the color scheme notwithstanding) because the story is quite simple.

The analysts divided the data into three time periods: 1980-94, 1995-15, 2016-23. The NBA seasons were summarized using a battery of 15 metrics arranged in a circle. In the first period, all but 3 of the metrics sat much above the average level (indicated by the inner circle). In the second period, all 15 metrics reduced below the average, and the third period is somewhat of a mirror image of the first, which is the main message.

***

The puzzle: why prefer this circular arrangement to a rectangular arrangement?

Here is what the same graph looks like in a rectangular arrangement:

One plausible justification for the circular arrangement is if the metrics can be clustered so that nearby metrics are semantically related.

Nevertheless, the same semantics appear in a rectangular arrangement. For example, P3-P3A are three point scores and attempts while P2-P2A are two-pointers. That is a key trend. They are neighborhoods in this arrangement just as they are in the circular arrangement.

So the real advantage is when the metrics have some kind of periodicity, and the wraparound point matters. Or, that the data are indexed to directions so north, east, south, west are meaningful concepts.

If you've found other use cases, feel free to comment below.

***

I can't end this post without returning to the colors. If one can take a negative image of the original chart, one should. Notice that the colors that dominate our attention - the yellow background, and the black lines - have no data in them: yellow being the canvass, and black being the gridlines. The data are found in the white polygons.

The other informative element, as one learns from the caption, is the "blue dashed line" that represents the value zero (i.e. average) in the standardized scale. Because the size of the image was small in the print magazine that I was reading, and they selected a dark blue encroaching on black, I had to squint hard to find the blue line.

##### Jun 17, 2024

This Financial Times report illustrates the reason why we should adjust data.

The story explores the trend in economic statistics during 14 years of governing by conservatives. One of those metrics is so-called council funding (local governments). The graphic is interactive: as the reader scrolls the page, the chart transforms.

The first chart shows the "raw" data.

The vertical axis shows year-on-year change in funding. It is an index relative to the level in 2010. From this line chart, one concludes that council funding decreased from 2010 to around 2016, then grew; by 2020, funding has recovered to the level of 2010 and then funding expanded rapidly in recent years.

When the reader scrolls down, this chart is replaced by another one:

This chart contains a completely different picture. The line dropped from 2010 to 2016 as before. Then, it went flat, and after 2021, it started raising, even though by 2024, the value is still 10 percent below the level in 2010.

What happened? The data journalist has taken the data from the first chart, and adjusted the values for inflation. Inflation was rampant in recent years, thus, some of the raw growth have been dampened. In economics, adjusting for inflation is also called expressing in "real terms". The adjustment is necessary because the same dollar (hmm, pound) is worth less when there is inflation. Therefore, even though on paper, council funding in 2024 is more than 25 percent higher than in 2010, inflation has gobbled up all of that and more, to the point in which, in real terms, council funding has fallen by 20 percent.

Wait, they have a third chart:

It's unfortunate they didn't stabilize the vertical scale. Relative to the middle chart, the lowest point in this third chart is about 5 percent lower, while the value in 2024 is about 10 percent lower.

This means, they performed a second adjustment - for population change. It is a simple adjustment of dividing by the population. The numbers look worse probably because population has grown during these years. Thus, even if the amount of funding stayed the same, the money would have to be split amongst more people. The per-capita adjustment makes this point clear.

***

The final story is much different from the initial one. Not only was the magnitude of change different but the direction of change reversed.

Whenever it comes to adjustments, remember that all adjustments are subjective. In fact, choosing not to adjust is also subjective. Not adjusting is usually much worse.

## Excess delay

##### Jun 05, 2024

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.

***

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.

***

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).

***

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.

## Prime visual story-telling

##### May 23, 2024

A story from the New York Times about New York City neighborhoods has been making the rounds on my Linkedin feed. The Linkedin post sends me to this interactive data visualization page (link).

Here, you will find a multi-colored map.

The colors show the extant of named neighborhoods in the city. If you look closely, the boundaries between neighborhoods are blurred since it's often not clear where one neighborhood ends and where another one begins. I was expecting this effect when I recognize the names of the authors, who have previously published other maps that obsess over spatial uncertainty.

I clicked on an area for which I know there may be differing opinions:

There was less controversy than I expected.

***

What was the dataset behind this dataviz project? How did they get such detailed data on every block of the city? Wouldn't they have to interview a lot of residents to compile the data?

I'm quite impressed with what they did. They put up a very simple survey (emphasis on: very simple). This survey is only possible with modern browser technology. It asks the respondent to pinpoint the location of where they live, and name their neighborhood. Then it asks the respondent to draw a polygon around their residence to include the extant of the named neighborhood. This consists of a few simple mouse clicks on the map that shows the road network. Finally, the survey collects optional information on alternative names for the neighborhood, etc.

When they process the data, they assign the respondent's neighborhood name to all blocks encircled by the polygon. This creates a lot of data in a few brush strokes, so to speak. This is a small (worthwhile) tradeoff even though the respondent didn't really give an answer for every block.

***

Bear with me, I'm getting to the gist of this blog post. The major achievement isn't the page that was linked to above. The best thing the dataviz team did here is the visual story that walks the reader through insights drawn from the dataviz. You can find the visual story here.

What are the components of a hugely impressive visual story?

• It combines data visualization with old-fashioned archival research. The historical tidbits add a lot of depth to the story.
• It combines data visualization with old-fashioned reporting. The quotations add context to how people think about neighborhoods - something that cannot be obtained from the arms-length process of conducting an online survey.
• It highlights curated insights from the underlying data - even walking the reader step by step through the relevant sections of the dataviz that illustrate these insights.

At the end of this story, some fraction of users may be tempted to go back to the interactive dataviz to search for other insights, or obtain answers to their personalized questions. They are much better prepared to do so, having just seen how to use the interactive tool!

***

The part of the visual story I like best is toward the end. Instead of plotting all the data on the map, they practice some restraint, and filter the data. They show the boundaries that have reached at least a certain level of consensus among the respondents.

The following screenshot shows those areas for which at least 90% agree.

Pardon the white text box, I wasn't able to remove it.

***

One last thing...

Every time an analyst touches data, or does something with data, s/he imposes assumptions, and sometimes, these assumptions are so subtle that even the analyst may not have noticed. Frequently, these assumptions are baked into the analytical "models," which is why they may fall through the cracks.

One such assumption in making this map is that every block in the city belongs to at least one named neighborhood. An alternative assumption is that neighborhoods are named only because certain blocks have things in common, and because these naming events occur spontaneously, it's perfectly ok to have blocks that aren't part of any named neighborhood.

## Chart without an axis

##### May 13, 2024

When it comes to global warming, most reports cite a single number such as an average temperature rise of Y degrees by year X. Most reports also claim the existence of a consensus within scientists. The Guardian presented the following chart that shows the spread of opinions amongst the experts.

Experts were asked how many degrees they expect average global temperature to increase by 2100. The estimates ranged from "below 1.5 degrees" to "5 degrees or more". The most popular answer was 2.5 degrees. Roughly three out of four respondents picked a number at 2.5 degrees or above. The distribution is close to symmetric around the middle.

***

What kind of chart is this?

It's a type of histogram, given that the horizontal axis shows binned ranges of temperature change while the vertical axis shows number of respondents (out of 380).

A (count) histogram typically encodes the count data in the vertical axis. Did you notice there isn't a vertical axis?

That's because the chart has an abnormal axis. Each of the 380 respondents is shown here as a cell. What looks like a "column" is actually two-dimensional. Each row of cells has 10 slots. To find out how many respondents chose the 2.5 celsius category, you count the number of rows and then the number of stray items on top. (It's 132.)

Only the top row of cells can be partially filled so the general shape of the distribution isn't affected much. However, the lack of axis labels makes it hard to learn the count of each column.

It's even harder to know the proportions of respondents, which should be the primary message of the chart. The proportion would have been possible to show if the maximum number of rows was set to 38. The maximum number of rows on the above chart is 22. Using 38 rows leads to a chart with a lot of white space as the tallest column (count of 132) is roughly 35% of the total response.

At the end, I'm not sure this variant of histogram beats the standard histogram.

## The curse of dimensions

##### Mar 20, 2024

Usually the curse of dimensions concerns data with many dimensions. But today I want to talk about a different kind of curse. This is the curse of dimensions in mapping.

We are only talking about a few dimensions, typically between 3 and 6, so small number of dimensions. And yet it's already a curse. Maps are typically drawn in two dimensions. Those two dimensions are usually spoken for: they show the x- and y-coordinate of space. If we want to include a third, fourth or fifth dimension of data on the map, we have to appeal to colors, shapes, and so on. Cartographers have long realized that adding dimensions involves tradeoffs.

***

Andrew featured some colored bubble maps in a recent post. Here is one example:

The above map shows the proportion of population in each U.S. county that is Hispanic. Each county is represented by a bubble pinned to the centroid of the county. The color of the bubble shows the data, divided into demi-deciles so they are using a equal-width binning method. The size of a bubble indicates the size of a county.

The map is sometimes called a "Dorling map" after its presumptive original designer.

I'm going to use this map to explore the curse of dimensions.

***

It's clear from the design that county-level details are regarded as extremely important. As there are about 3,000 counties in the U.S., I don't see how any visual design can satisfy this requirement without giving up clarity.

More details require more objects, which spread readers' attention. More details contain more stories, but that too dilutes their focus.

Another principle of this map is to not allow bubbles to overlap. Of course, having bubbles overlap or print on top of one another is a visual faux pas. But to prevent such behavior on this particular design means the precise locations are sacrificed. Consider the eastern seaboard where there are densely populated counties: they are not pinned to their centroids. Instead, the counties are pushed out of their normal positions, similar to making a cartogram.

I remarked at the start – erroneously but deliberately – that each bubble is centered at the centroid of each county. I wonder how many of you noticed the inaccuracy of that statement. If that rule were followed, then the bubbles in New England would have overlapped and overprinted.

This tradeoff affects how we perceive regional patterns, as all the densely populated regions are bent out of shape.

Another aspect of the data that the designer treats as important is county population, or rather relative county population. Relative – because bubble size don't portray absolutes, plus the designer didn't bother to provide a legend to decipher bubble sizes.

The tradeoff is location. The varying bubble sizes, coupled with the previous stipulation of no overlapping, push bubbles from their proper centroids. This forced displacement disproportionately affects larger counties.

***

What if we are willing to sacrifice county-level details?

In this setting, we are not obliged to show every single county. One alternative is to perform spatial smoothing. Intuitively, think about the following steps: plot all these bubbles in their precise locations, turn the colors slightly transparent, let them overlap, blend away the edges, and then we have a nice picture of where the Hispanic people are located.

I have sacrificed the county-level details but the regional pattern becomes much clearer, and we don't need to deviate from the well-understood shape of the standard map.

This version reminds me of the language maps that Josh Katz made.

Here is an old post about these maps.

This map design only reduces but does not eliminate the geographical inaccuracy. It uses the same trick as the Dorling map: the "vertical" density of population has been turned into "horizontal" span. It's a bit better because the centroids are not displaced.

***

Which map is better depends on what tradeoffs one is making. In the above example, I'd have made different choices.

One final thing – it's minor but maybe not so minor. Most of the bubbles on the map especially in the middle are tiny; as most of them have Hispanic proportions that are on the left side of the scale, they should be showing light orange. However, all of them appear darker than they ought to be. That's because each bubble has a dark border. For small bubbles, the ratio of ink on the border is a high proportion of the ink for the entire object.

## My talk next week on histograms

##### Mar 08, 2024

Next Thursday (March 14), I'll be presenting at the Data Visualization New York Meetup, hosted by Naomi and Cameron. The event is in-person at Datadog's office. You can reserve your spot here.

This talk is brand new, based on some work inspired by a blog post by Andrew Gelman. One of Andrew's correspondents asked about a particular type of histogram. While exploring this topic, I filled some of my own gaps in knowledge about this deceptively simple chart form. I'll be sharing this story.

Bits and pieces have appeared before on my blog. See this, this, and this for background.

If you're attending the talk, come up and say hi.

## Do you want a taste of the new hurricane cone?

##### Mar 05, 2024

The National Hurricane Center (NHC) put out a press release (link to PDF) to announce upcoming changes (in August 2024) to their "hurricane cone" map. This news was picked up by Miami Herald (link).

The above example is what the map looks like. (The data are probably fake since the new map is not yet implemented.)

The cone map has been a focus of research because experts like Alberto Cairo have been highly critical of its potential to mislead. Unfortunately, the more attention paid to it, the more complicated the map has become.

The bottom layer is the so-called "cone". This is the white patch labeled below as the "potential track area (day 1-5)".  Researchers dislike this element because they say readers tend to misinterpret the cone as predicting which areas would be damaged by hurricane winds when the cone is intended to depict the uncertainty about the path of the hurricane. Prior criticism has led the NHC to add the text at the top of the chart, saying "The cone contains the probable path of the storm center but does not show the size of the storm. Hazardous conditions can occur outside of the cone."

The middle layer are the multi-colored bits. Two of these show the areas for which the NHC has issued "watches" and "warnings". All of these color categories represent wind speeds at different times. Watches and warnings are forecasts while the other colors indicate "current" wind speeds.

The top layer consists of black dots. These provide a single forecast of the most likely position of the storm, with the S, H, M labels indicating the most likely range of wind speeds at forecast times.

***

Let's compare the new cone map to a real hurricane map from 2020. (This older map came from a prior piece also by NHC.)

Can we spot the differences?

To my surprise, the differences were minor, in spite of the pre-announced changes.

The first difference is a simplification. Instead of dividing the white cone (the bottom layer) into two patches -- a white patch for days 1-3, and a dotted transparent patch for days 4-5, the new map aggregates the two periods. Visually, simplifying makes the map less busy but loses the implicit acknowledge found in the old map that forecasts further out are not as reliable.

The second point of departure is the addition of "inland" warnings and watches. Notice how the red and blue areas on the old map hugged the coastline while the red and blue areas on the new map reach inland.

Both changes push the bottom layer, i.e. the cone, deeper into the background. It's like a shrink-flation ice cream cone that has a tiny bit of ice cream stuffed deep in its base.

***

How might one improve the cone map? I'd start by dismantling the layers. The three layers present answers to different problems, albeit connected.

Let's begin with the hurricane forecasting problem. We have the current location of the storm, and current measurements of wind speeds around its center. As a first requirement, a forecasting model predicts the path of the storm in the near future. At any time, the storm isn't a point in space but a "cloud" around a center. The path of the storm traces how that cloud will move, including any expansion or contraction of its radius.

That's saying a lot. To start with, a forecasting model issues the predicted average path -- the expected path of the storm's center. This path is (not competently) indicated by the black dots in the top layer of the cone map. These dots offer only a sampled view of the average path.

Not surprisingly, there is quite a bit of uncertainty about the future path of any storm. Many models simulate future worlds, generating many predictions of the average paths. The envelope of the most probable set of paths is the "cone". The expanding width of the cone over time reflects the higher uncertainty of our predictions further into the future. Confusingly, this cone expansion does not depict spatial expansion of either the storm's size or the potential areas that may suffer the greatest damage. Both of those tend to shrink as hurricanes move inland.

Nevertheless, the cone and the black dots are connected. The path drawn out by the black dots should be the average path of the center of the storm.

The forecasting model also generates estimates of wind speeds. Those are given as labels inside the black dots. The cone itself offers no information about wind speeds. The map portrays the uncertainty of the position of the storm's center but omits the uncertainty of the projected wind speeds.

The middle layer of colored patches also inform readers about model projections - but in an interpreted manner. The colors portray hurricane warnings and watches for specific areas, which are based on projected wind speeds from the same forecasting models described above. The colors represent NHC's interpretation of these model outputs. Each warning or watch simultaneously uses information on location, wind speed and time. The uncertainty of the projected values is suppressed.

I think it's better to use two focused maps instead of having one that captures a bit of this and a bit of that.

One map can present the interpreted data, and show the areas that have current warnings and watches. This map is about projected wind strength in the next 1-3 days. It isn't about the center of the storm, or its projected path. Uncertainty can be added by varying the tint of the colors, reflecting the confidence of the model's prediction.

Another map can show the projected path of the center of the storm, plus the cone of uncertainty around that expected path. I'd like to bring more attention to the times of forecasting, perhaps shading the cone day by day, if the underlying model has this level of precision.

***

Back in 2019, I wrote a pretty long post about these cone maps. Well worth revisiting today!

## Lost in the middle class

##### Feb 20, 2024

Washington Post asks people what it means to be middle class in the U.S. (link; paywall)

The following graphic illustrates one type of definition, purely based on income ranges.

For me, this chart is more taxing to read than it appears.

It can be read column by column. Each column represents a hypotheticial annual income for a family of four. People are asked whether they consider that family lower/working class, middle class or upper class. Be careful as the increments from column to column are not uniform.

Now, what's the question again? We're primarily interested in what incomes constitute middle class.

So, we should be looking at the deep green blocks that hang in the middle of each column. It's not easy to read the proportion of middle blocks in a stacked column chart.

***

I tried separating out the three perceived income classes, using a small-multiples design.

One can more directly see what income ranges are most popularly perceived as being in each income class.

***

The article also goes into alternative definitions of middle class, using more qualitative metrics, such as "able to pay all bills on time without worry". That's a whole other post.

## Neither the forest nor the trees

##### Feb 15, 2024

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.

***

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.

***

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.