If you blink, you'd miss this axis trick

When I set out to write this post, I was intending to make a quick point about the following chart found in the current issue of Harvard Magazine (link):

This chart concerns the "tectonic shift" of undergraduates to STEM majors at the expense of humanities in the last 10 years.

I like the chart. The dot plot is great for showing this data. They placed the long text horizontally. The use of color is crucial, allowing us to visually separate the STEM majors from the humanities majors.

My intended post is to suggest dividing the chart into four horizontal slices, each showing one of the general fields. It's a small change that makes the chart even more readable. (It has the added benefit of not needing a legend box.)

***

Then, the axis announced itself.

I was baffled, then disgusted.

Here is a magnified view of the axis:

It's not a linear scale, as one would have expected. What kind of transformation did they use? It's baffling.

Notice the following features of this transformed scale:

• It can't be a log scale because many of the growth values are negative.
• The interval for 0%-25% is longer than for 25%-50%. The interval for 0%-50% is also longer than for 50%-100%. On the positive side, the larger values are pulled in and the smaller values are pushed out.
• The interval for -20%-0% is the same length as that for 0%-25%. So, the transformation is not symmetric around 0

I have no idea what transformation was applied. I took the growth values, measured the locations of the dots, and asked Excel to fit a polynomial function, and it gave me a quadratic fit, R square > 99%.

This formula fits the values within the range extremely well. I hope this isn't the actual transformation. That would be disgusting. Regardless, they ought to have advised readers of their unusual scale.

***

Without having the fitted formula, there is no way to retrieve the actual growth values except for those that happen to fall on the vertical gridlines. Using the inverse of the quadratic formula, I deduced what the actual values were. The hardest one is for Computer Science, since the dot sits to the right of the last gridline. I checked that value against IPEDS data.

The growth values are not extreme, falling between -50% and 125%. There is nothing to be gained by transforming the scale.

The following chart undoes the transformation, and groups the majors by field as indicated above:

***

Yesterday, I published a version of this post at Andrew's blog. Several readers there figured out that the scale is the log of the relative ratio of the number of degrees granted. In the above notation, it is log10(100%+x), where x is the percent change in number of degrees between 2011 and 2021.

Here is a side-by-side view of the two scales:

The chart on the right spreads the negative growth values further apart while slightly compressing the large positive values. I still don't think there is much benefit to transforming this set of data.

 

P.S. [1/31/2023]

(1) A reader on Andrew's blog asked what's wrong with using the log relative ratio scale. What's wrong is exactly what this post is about. For any non-linear scale, the reader can't make out the values between gridlines. In the original chart, there are four points that exist between 0% and 25%. What values are those? That chart is even harder because now that we know what the transform is, we'd need to first think in terms of relative ratios, so 1.25 instead of 25%, then think in terms of log, if we want to know what those values are.

(2) The log scale used for change values is often said to have the advantage that equal distances on either side represent counterbalancing values. For example, (1.5) (0.66) = (3/2) (2/3)  = 1. But this is a very specific scenario that doesn't actually apply to our dataset.  Consider these scenarios:

History: # degrees went from 1000 to 666 i.e. Relative ratio = 2/3
Psychology: # degrees went from 2000 to 3000 i.e. Relative ratio = 3/2

The # of History degrees dropped by 334 while the number of Psychology degrees grew by 1000 (Psychology I think is the more popular major)

History: # degrees went from 1000 to 666 i.e. Relative ratio = 2/3
Psychology: from 1000 to 1500, i.e. Relative ratio = 3/2

The # of History degrees dropped by 334 while # of Psychology degrees grew by 500
(Assume same starting values)

History: # degrees went from 1000 to 666 i.e. Relative ratio = 2/3
Psychology: from 666 to 666*3/2 = 999 i.e. Relative ratio = 3/2

The # of History degrees dropped by 334 while # of Psychology degrees grew by 333
(Assume Psychology's starting value to be History's ending value)

Psychology: # degrees went from 1000 to 1500 i.e. Relative ratio = 3/2
History: # degrees went from 1500 to 1000 i.e. Relative ratio = 2/3

The # of Psychology degrees grew by 500 while the # of History degrees dropped by 500
(Assume History's starting value to be Psychology's ending value)

 

 

Energy efficiency deserves visual efficiency

Long-time contributor Aleksander B. found a good one, in the World Energy Outlook Report, published by IEA (International Energy Agency).

The use of balloons is unusual, although after five minutes, I decided I must do some research to have any hope of understanding this data visualization.

A lot is going on. Below, I trace my own journey through this chart.

The text on the top left explains that the chart concerns emissions and temperature change. The first set of balloons (the grey ones) includes helpful annotations. The left-right position of the balloons indicates time points, in 10-year intervals except for the first.

The trapezoid that sits below the four balloons is more mysterious. It's labelled "median temperature rise in 2100". I debate two possibilities: (a) this trapezoid may serve as the fifth balloon, extending the time series from 2050 to 2100. This interpretation raises a couple of questions: why does the symbol change from balloon to trapezoid? why is the left-right time scale broken? (b) this trapezoid may represent something unrelated to the balloons. This interpretation also raises questions: its position on the horizontal axis still breaks the time series; and  if the new variable is "median temperature rise", then what determines its location on the chart?

That last question is answered if I move my glance all the way to the right edge of the chart where there are vertical axis labels. This axis is untitled but the labels shown in degree Celsius units are appropriate for "median temperature rise".

Turning to the balloons, I wonder what the scale is for the encoded emissions data. This is also puzzling because only a few balloons wear data labels, and a scale is nowhere to be found.

The gridlines suggests that the vertical location of the balloons is meaningful. Tracing those gridlines to the right edge leads me back to the Celsius scale, which seems unrelated to emissions. The amount of emissions is probably encoded in the sizes of the balloons although none of these four balloons have any data labels so I'm rather flustered. My attention shifts to the colored balloons, a few of which are labelled. This confirms that the size of the balloons indeed measures the amount of emissions. Nevertheless, it is still impossible to gauge the change in emissions for the 10-year periods.

The colored balloons rising above, way above, the gridlines is an indication that the gridlines may lack a relationship with the balloons. But in some charts, the designer may deliberately use this device to draw attention to outlier values.

Next, I attempt to divine the informational content of the balloon strings. Presumably, the chart is concerned with drawing the correlation between emissions and temperature rise. Here I'm also stumped.

I start to look at the colored balloons. I've figured out that the amount of emissions is shown by the balloon size but I am still unclear about the elevation of the balloons. The vertical locations of these balloons change over time, hinting that they are data-driven. Yet, there is no axis, gridline, or data label that provides a key to its meaning.

Now I focus my attention on the trapezoids. I notice the labels "NZE", "APS", etc. The red section says "Pre-Paris Agreement" which would indicate these sections denote periods of time. However, I also understand the left-right positions of same-color balloons to indicate time progression. I'm completely lost. Understanding these labels is crucial to understanding the color scheme. Clearly, I have to read the report itself to decipher these acronyms.

The research reveals that NZE means "net zero emissions", which is a forecasting scenario - an utterly unrealistic one - in which every country is assumed to fulfil fully its obligations, a sort of best-case scenario but an unattainable optimum. APS and STEPS embed different assumptions about the level of effort countries would spend on reducing emissions and tackling global warming.

At this stage, I come upon another discovery. The grey section is missing any acronym labels. It's actually the legend of the chart. The balloon sizes, elevations, and left-right positions in the grey section are all arbitrary, and do not represent any real data! Surprisingly, this legend does not contain any numbers so it does not satisfy one of the traditional functions of a legend, which is to provide a scale.

There is still one final itch. Take a look at the green section:

What is this, hmm, caret symbol? It's labeled "Net Zero". Based on what I have been able to learn so far, I associate "net zero" to no "emissions" (this suggests they are talking about net emissions not gross emissions). For some reason, I also want to associate it with zero temperature rise. But this is not to be. The "net zero" line pins the balloon strings to a level of roughly 2.5 Celsius rise in temperature.

Wait, that's a misreading of the chart because the projected net temperature increase is found inside the trapezoid, meaning at "net zero", the scientists expect an increase in 1.5 degrees Celsius. If I accept this, I come face to face with the problem raised above: what is the meaning of the vertical positioning of the balloons? There must be a reason why the balloon strings are pinned at 2.5 degrees. I just have no idea why.

I'm also stealthily presuming that the top and bottom edges of the trapezoids represent confidence intervals around the median temperature rise values. The height of each trapezoid appears identical so I'm not sure.

I have just learned something else about this chart. The green "caret" must have been conceived as a fully deflated balloon since it represents the value zero. Its existence exposes two limitations imposed by the chosen visual design. Bubbles/circles should not be used when the value of zero holds significance. Besides, the use of balloon strings to indicate four discrete time points breaks down when there is a scenario which involves only three buoyant balloons.

***

The underlying dataset has five values (four emissions, one temperature rise) for four forecasting scenarios. It's taken a lot more time to explain the data visualization than to just show readers those 20 numbers. That's not good!

I'm sure the designer did not set out to confuse. I think what happened might be that the design wasn't shown to potential readers for feedback. Perhaps they were shown only to insiders who bring their domain knowledge. Insiders most likely would not have as much difficulty with reading this chart as did I.

This is an important lesson for using data visualization as a means of communications to the public. It's easy for specialists to assume knowledge that readers won't have.

For the IEA chart, here is a list of things not found explicitly on the chart that readers have to know in order to understand it.

• Readers have to know about the various forecasting scenarios, and their acronyms (APS, NZE, etc.). This allows them to interpret the colors and section titles on the chart, and to decide whether the grey section is missing a scenario label, or is a legend.
• Since the legend does not contain any scale information, neither for the balloon sizes nor for the temperatures, readers have to figure out the scales on their own. For temperature, they first learn from the legend that the temperature rise information is encoded in the trapezoid, then find the vertical axis on the right edge, notice that this axis has degree Celsius units, and recognize that the Celsius scale is appropriate for measuring median temperature rise.
• For the balloon size scale, readers must resist the distracting gridlines around the grey balloons in the legend, notice the several data labels attached to the colored balloons, and accept that the designer has opted not to provide a proper size scale.

Finally, I still have several unresolved questions:

• The horizontal axis may have no meaning at all, or it may only have meaning for emissions data but not for temperature
• The vertical positioning of balloons probably has significance, or maybe it doesn't
• The height of the trapezoids probably has significance, or maybe it doesn't

 

 

Speedometer charts: love or hate

Pie chart hate is tired. In this post, I explain my speedometer hate. (Also called gauges,  dials)

Next to pie charts, speedometers are perhaps the second most beloved chart species found on business dashboards. Here is a typical example:

 

For this post, I found one on Reuters about natural gas in Europe. (Thanks to long-time contributor Antonio R. for the tip.)

The reason for my dislike is the inefficiency of this chart form. In classic Tufte-speak, the speedometer chart has a very poor data-to-ink ratio. The entire chart above contains just one datum (73%). Most of the ink are spilled over non-data things.

This single number has a large entourage:

- the curved axis
- ticks on the axis
- labels on the scale
- the dial
- the color segments
- the reference level "EU target"

These are not mere decorations. Taking these elements away makes it harder to understand what's on the chart.

Here is the chart without the curved axis:

Here is the chart without axis labels:

Here is the chart without ticks:

When the tick labels are present, the chart still functions.

Here is the chart without the dial:

The datum is redundantly encoded in the color segments of the "axis".

Here is the chart without the dial or the color segments:

If you find yourself stealing a peek at the chart title below, you're not alone.

All versions except one increases our cognitive load. This means the entourage is largely necessary if one encodes the single number in a speedometer chart.

The problem with the entourage is that readers may resort to reading the text rather than the chart.

***

The following is a minimalist version of the Reuters chart:

I removed the axis labels and the color segments. The number 73% is shown using the dial angle.

The next chart adds back the secondary message about the EU target, as an axis label, and uses color segments to show the 73% number.

Like pie charts, there are limited situations in which speedometer charts are acceptable. But most of the ones we see out there are just not right.

***

One acceptable situation is to illustrate percentages or proportions, which is what the EU gas chart does. Of course, in that situation, one can alo use a pie chart without shame.

For illustrating proportions, I prefer to use a full semicircle, instead of the circular sector of arbitrary angle as Reuters did. The semicircle lends itself to easy marks of 25%, 50%, 75%, etc, eliminating the need to print those tick labels.

***

One use case to avoid is numeric data.

Take the regional sales chart pulled randomly from a Web search above:

These charts are completely useless without the axis labels.

Besides, because the span of the axis isn't 0% to 100%, every tick mark must be labelled with the numeric value. That's a lot of extra ink used to display a single value!

Who trades with Sweden

It's great that the UN is publishing dataviz but it can do better than this effort:

Certain problems are obvious. The country names turned sideways. The meaningless use of color. The inexplicable sequencing of the country/region.

Some problems are subtler. "Area, nes" - upon research - is a custom term used by UN Trade Statistics, meaning "not elsewhere specified".

The gridlines are debatable. Their function is to help readers figure out the data values if they care. The design omitted the top and bottom gridlines, which makes it hard to judge the values for USA (dark blue), Netherlands (orange), and Germany (gray).

See here, where I added the top gridline.

Now, we can see this value is around 3.6, just over the halfway point between gridlines.

***

A central feature of trading statistics is "balance". The following chart makes it clear that the positive numbers outweigh the negative numbers in the above chart.

At the time I made the chart, I wasn't sure how to interpret the gap of 1.3%. Looking at the chart again, I think it's saying Sweden has a trade surplus equal to that amount.

Superb tile map offering multiple avenues for exploration

Here's a beauty by WSJ Graphics:

The article is here.

This data graphic illustrates the power of the visual medium. The underlying dataset is complex: power production by type of source by state by month by year. That's more than 90,000 numbers. They all reside on this graphic.

Readers amazingly make sense of all these numbers without much effort.

It starts with the summary chart on top.

The designer made decisions. The data are presented in relative terms, as proportion of total power production. Only the first and last years are labeled, thus drawing our attention to the long-term trend. The order of the color blocks is carefully selected so that the cleaner sources are listed at the top and the dirtier sources at the bottom. The order of the legend labels mirrors the color blocks in the area chart.

It takes only a few seconds to learn that U.S. power production has largely shifted away from coal with most of it substituted by natural gas. Other than wind, the green sources of power have not gained much ground during these years - in a relative sense.

This summary chart serves as a reading guide for the rest of the chart, which is a tile map of all fifty states. Embedded in the tile map is a small-multiples arrangement.

***

The map offers multiple avenues for exploration.

Some readers may look at specific states. For example, California.

Currently, about half of the power production in California come from natural gas. Notably, there is no coal at all in any of these years. In addition to wind, solar energy has also gained. All of these insights come without the need for any labels or gridlines!

Browsing around California, readers find different patterns in other Western states like Oregon and Washington.

Hydroelectric energy is the dominant source in those two states, with wind gradually taking share.

At this point, readers realize that the summary chart up top hides remarkable state-level variations.

***

There are other paths through the map.

Some readers may scan the whole map, seeking patterns that pop out.

One such pattern is the cluster of states that use coal. In most of these states, the proportion of coal has declined.

Yet another path exists for those interested in specific sources of power.

For example, the trend in nuclear power usage is easily followed by tracking the purple. South Carolina, Illinois and New Hampshire are three states that rely on nuclear for more than half of its power.

I wonder what happened in Vermont about 8 years ago.

The chart says they renounced nuclear energy. Here is some history. This one-time event caused a disruption in the time series, unique on the entire map.

***

This work is wonderful. Enjoy it!

Start at zero, or start at wherever

Andrew's post about start-at-zero helps me refine my own thinking on this evergreen topic.

The specific example he gave is this one:

The dataset is a numeric variable (y) with values over time (x). The minimum numeric value is around 3 and the range of values is from around 3 to just above 20. His advice is "If zero is in the neighborhood, invite it in". (Link)

The rule, as usual, sounds simpler than it really is. In the discussion, Andrew highlights several considerations.

Is zero a meaningful reference value? In his example, we assume it is and so we invite zero in. But, as Andrew also says, if zero is meaningless, then recall the invitation. So context must be accounted for.

In Chapter 1 of Numbersense (link), I looked at some SAT score data of applicants to competitive colleges. Is zero a meaningful reference value for SAT scores? Someone might argue yes, since it is the theoretical minimum score that anyone could get from the test. Any statistician will likely say no, since a competitive college will have never seen an applicant submitting a score of zero, or anywhere close to zero. Thus, starting such a chart at zero inserts a lot of whitespace and draws attention to a useless insight - how far above the theoretical worst performer is someone's score.

***

What about the left panel of Andrew's chart makes us uncomfortable? I ask myself this question. My answer is that the horizontal axis highlights an arbitrary value that distracts from the key patterns of the data.

As shown below, the arbitrary value is ~2.5. This is utterly meaningless.

What if 0 is also a meaningless value for this dataset? I'd recommend "bench the axis". Like this:

An axis is a tool to help readers understand a chart. If it isn't serving a function, an axis doesn't need to be there. When I choose a line chart for time-series data, I'm drawing attention to temporal change in the numeric values, or the range of values. I'm not saying something about the values relative to some reference number.

From this example, we also see that the horizontal axis should not be regarded as a hanger for time labels. Time labels can exist by themselves.

 

 

Speaking to the choir

A friend found the following chart about the "carbon cycle", and sent me an exasperated note, having given up on figuring it out. The chart came from a report, and was reprinted in Ars Technica (link).

The problem with the chart is that the designer is speaking to the choir. One must know a lot about the carbon cycle already to make sense of everything that's going on.

We see big and small arrows pointing up or down. Each arrow has a number attached to it, plus a range inside brackets. These numbers have no units, and it's not obvious what they are measuring.

The arrows come in a variety of colors. The colors are explained by labels but the labels dexcribe apparently unrelated concepts (e.g. fossil CO2 and land-use change).

Interspersed with the arrows is a singular dot. The dot also has a number attached to it. The number wears a plus sign, which signals it's being treated differently than the quantities with up arrows.

The singular dot is an outcast, ostracized from the community of dots in the bottom part of the chart. These dots have labels but no numbers. They come in different sizes but no scale is provided.

The background is divided into three parts, showing the atmosphere, the land mass, and the ocean. The placement of the arrows and dots suggests each measured quantity concerns one of these three parts. Well... except the dot labeled "surface sediments" that sit on the boundary of the land mass and the ocean.

The three-way classification is only one layer of the chart. A different classification is embedded in the color scheme. The gray, light green, and aquamarine arrows in the sky find their counterparts in the dots of the land mass, and the ocean.

What's more, the boundaries between land and sky, and between land and ocean are also painted with those colors. These boundary segments have been given different colors so that the lengths of these segments seem to contain data but we aren't sure what.

At this point, I noticed thin arrows which appear to depict back and forth flows. There may be two types of such exchanges, one indicated by a cycle, the other by two straight arrows in opposite directions. The cycles have no numbers while each pair of straight thin arrows gets two numbers, always identical.

At the bottom of the chart is a annotation in red: "Budget imbalance = -1.0". Presumably some formula ties the numbers shown above to this -1.0 result. We still don't know the units, and it's unclear if -1.0 is a bad number. A negative number shown in red typically indicates a bad number but how bad is it?

Finally, on the top right corner, I found a legend. It's not obvious at first because the legend symbols (arrows and dots) are shown in gray, a color not used elsewhere on the chart. It appears as if it represents another color category. The legend labels do little for me. What is an "anthropogenic flux"? What does the unit of "GtCO2" stand for? Other jargon includes "carbon cycling" and "stocks". The entire diagram is titled "carbon cycle" while the "carbon cycling" thin arrows are only a small part of the diagram.

The bottom line is I have no idea what this chart is saying to me, other than that the earth is a complex system, and that the designer has tried valiantly to impregnate the diagram with lots of information. If I am well read in environmental science, my experience is likely different.

 

 

 

 

 

Surging gas prices

A reader finds this chart hard to parse:

The chart shows the trend in gas prices in New York in the past two years.

This is a case in which the simple line chart works very well.

I added annotations as the reasons behind the decline and rise in prices are reasonably clear. 

One should be careful when formatting dates. The legend of the original chart looks like this:

In the U.S., dates typically use a M/D/Y format. The above dates are ambiguous. "Aug 19" can be August 19th or August, xx19.

Working hard at clarity

As I am preparing another blog post about the pandemic, I came across the following data graphic, recently produced by the CDC for a vaccine advisory board meeting:

This is not an example of effective visual communications.

***

For one thing, readers are directed to scour the footnotes to figure out what's going on. If we ignore those for the moment, we see clusters of bubbles that have remained pretty stable from December 2020 to August 2021. The data concern some measure of Americans' intent to take the COVID-19 vaccine. That much we know.

There may have been a bit of an upward trend between January and May, although if you were shown the clusters for December, February and April, you'd think the trend's been pretty flat. 

***

But those colors? What could they represent? You'd surely have to fish this one out of the footnotes. Specifically, this obtuse sentence: "Surveys with multiple time points are shown with the same color bubble for each time point." I had to read it several times. I think it simply means "Color represents the pollster." 

Then it adds: "Surveys with only one time point are shown in gray." which simply means "All pollsters who have only one entry in the dataset are grouped together and shown in gray."

Another problem with this chart is over-plotting. Look at the July cluster. It's impossible to tell how many polls were conducted in July because the circles pile on top of one another. 

***

The appearance of the flat trend is a result of two unfortunate decisions made by the designer. If I retained the chart form, I'd have produced something that looks like this:

The first design choice is to expand the vertical axis to range from 0% to 100%. This effectively squeezes all the bubbles into a small range.

The second design choice is to enlarge the bubbles causing copious amount of overlapping. 

In particular, this decision blows up the Pew poll (big pink bubble) that contained 10 times the sample size of most of the other polls. The Pew outcome actually came in at 70% but the top of the pink bubble extends to over 80%. Because of this, the outlier poll of December 2020 - which surprisingly printed the highest number of all polls in the entire time window - no longer looks special. 

***

Now, let's see what else we can do to enhance this chart. 

I don't like how bubble size is used to encode the sample size. It creates a weird sensation for anyone who's familiar with sampling errors, and confidence regions. The Pew poll with 10 times the sample size is the most reliable poll of them all. Reliability means the error bars around the Pew poll outcome is the smallest of them all. I tend to think of the area around a point estimate as showing the sampling error so the Pew poll would be a dot, showing the high precision of that estimate. 

But that won't work because larger bubbles catch more of the reader's attention. So, in the following version, all dots have the same size. I encode reliability in the opacity of the color. The darker dots are polls that are more reliable, that have larger sample sizes.

Two of the pollsters have more frequent polling than others. In this next version, I highlighted those two, which reveals the trend better.

 

 

 

Metaphors, maps, and communicating data

There are some data visualization that are obviously bad. But what makes them bad?

Here is an example of such an effort:

This visualization of carbon emissions is not successful. There is precious little that a reader can learn from this chart without expensing a lot of effort. It's relatively easy to identify the largest emitters of carbon but since the data are not expressed per-capita, the chart mainly informs us which countries have the largest populations. 

The color of the bubbles informs readers which countries belong to which parts of the world. However, it distorts the location of countries within regions, and regions relative to regions, as the primary constraint is fitting the bubbles inside the shape of a foot.

The visualization gives a very rough estimate of the relative sizes of total emissions. The circles not being perfect circles don't help. 

It's relatively easy to list the top emitters in each region but it's hard to list the top 10 emitters in the world (try!) 

The small emitters stole all of the attention as they account for most of the labels - and they engender a huge web of guiding lines - an unsightly nuisance.

The diagram clings dearly to the "carbon footprint" metaphor. Does this metaphor help readers consume the emissions data? Conversely, does it slow them down?

A more conventional design uses a cartogram, a type of map in which the positioning of countries are roughly preserved while the geographical areas are coded to the data. Here's how it looks:

I can't seem to source this effort. If any reader can find the original source, please comment below.

This cartogram is a rearrangement of the footprint illustration. The map construct eliminates the need to include a color legend which just tells people which country is in which continent. The details of smaller countries are pushed to the bottom. 

In the footprint visualization, I'd even consider getting rid of the legend completely. This means trusting that readers know South Africa is part of Africa, and China is part of Asia.

Imagine: what if this chart comes without a color legend? Do we really need it?

***

I'd like to try a word cloud visual for this dataset. Something that looks like this (obviously with the right data encoding):

(This map is by Michael Tompsett who sells it here.)