Logging a sleight of hand

Andrew puts up an interesting chart submitted by one of his readers (link):

Bruce Knuteson who created this chart is pursuing a theory that there is some fishy going on in the stock markets over night (i.e. between the close of one day and the open of the next day). He split the price data into two interleaving parts: the blue line represents returns overnight and the green line represents returns intraday (from open of one day to the close of the same day). In this example related to Tesla's stock, the overnight "return" is an eyepopping 36850% while the intraday "return" is -46%.

This is an example of an average masking interesting details in the data. One typically looks at the entire sequence of values at once, while this analysis breaks it up into two subsequences. I'll write more about the data analysis at a later point. This post will be purely about the visualization.

***

It turns out that while the chart looks like a standard time series, it isn't. Bruce wrote out the following essential explanation:

The chart can't be interpreted without first reading this note.

The left chart (a) is the standard time-series chart we're thinking about. It plots the relative cumulative percentage change in the value of the investment over time. Imagine one buys $1 of Apple stock on day 1. It shows the cumulative return on day X, expressed as a percent relative to the initial investment amount. As mentioned above, the data series was split into two: the intraday return series (green) is dwarfed by the overnight return series (blue), and is barely visiable hugging the horizontal axis.

Almost without thinking, a graphics designer applies a log transform to the vertical axis. This has the effect of "taming" the extreme values in the blue line. This is the key design change in the middle chart (b). The other change is to switch back to absolute values. The day 1 number is now $1 so the day X number shows the cumulative value of the investment on day X if one started with $1 on day 1.

There's a reason why I emphasized the log transform over the switch to absolute values. That's because the relationship between absolute and relative values here is a linear one. If y(t) is the absolute cumulative value of $1 at time t, then the percent change r(t) = 100(y(t) -1). (Note that y(0) = 1 by definition.)  The shape of the middle chart is primarily conditioned by the log transform.

In the right chart (c), which is the design that Bruce features in all his work, the visual elements of chart (b) are retained while he replaced the vertical axis labels with those from chart (a). In other words, the lines show the cumulative absolute values while the labels show the relative cumulative percent returns.

I left this note on Gelman's blog (corrected a mislabeling of the chart indices):

I'm interested in the the sleight of hand related to the plots, also tying this back to the recent post about log scales. In plot (b) (a) [middle of the panel], he transformed the data to show the cumulative value of the investment assuming one puts $1 in the stock on day 1. He applied a log scale on the vertical axis. This is fine. Then in plot (c) (b), he retained the chart but changed the vertical axis labels so instead of absolute value of the investment, he shows percent changes relative to the initial value.

Why didn't he just plot the relative percent changes? Let y(t) be the absolute values and r(t) = the percent change = 100*(y(t) -1) is a simple linear transformation of y(t). This is where the log transform creates problems! The y(t) series is guaranteed to be positive since hitting y(t) = 0 means the entire investment is lost. However, the r(t) series can hit negative values and also cross over zero many times over time. Thus, log r(t) is inoperable. The problem is using the log transform for data that are not always positive, and the sleight of hand does not fix it!

Just pick any day in which the absolute return fell below $1, e.g. the last day of the plot in which the absolute value of the investment was down to $0.80. In the middle plot (b), the value depicted is ln(0.8) = -0.22. Note that the plot is in log scale, so what is labeled as $1 is really ln(1) = 0. If we instead try to plot the relative percent changes, then the day 1 number should be ln(0) which is undefined while the last number should be ln(-20%) which is also undefined.

This is another example of something umcomfortable about using log scales which I pointed out in this post. It's this idea that when we do log plots, we can freely substitute axis labels which are not directly proportional to the actual labels. It's plotting one thing, and labelling it something else. These labels are then disconnected from the visual encoding. It's against the goal of visualizing data.

 

Making major things easy, and minor things hard

A recent issue of Significance magazine carried the following stacked column chart showing how the driver license status of men and women change as they age. The data came from the U.K.

Quick question - what percentage of British men in their sixties hold full driver licenses?

***

I was just kidding. Those questions can't be quickly answered on a stacked column chart. That's because you have to find the axis, and then mentally invert the axis.

On that chart, larger values are shown pointing down (green) and also pointing up (blue), and ... well, I don't have words for the yellow. In fact, the yellow segments, showing people without licenses, are possibly the most important category for this report.

In making decisions about visualizing data, it's important to separate out the major things from the minor things.

***

Here is a reimagination of the chart using connected dots:

What is hard to do using this chart is to verify that the three proportions add to 100%. What is easy is to read off the proportion for any gender, age and license status subgroup.

It's really quite intricate how these researchers binned the age data. There are bins of size 1, 4, 5 and 10, plus the top group is 85 and above. The way I handled these is to turn everything to 1-year bins. I assume that in the wider bins, we don't have precise data for each age, and the bin value is the average among the bin, thus it is as if someone had drawn a horizontal line across the bin width. (I left the top bin alone as I don't know what is the maximum age of a person in this study.)

***

Those of you who have laminated the flowchart of data visualization are probably irate. According to such a flowchart, one must use a column chart because the x variable (age band) has irregularly-sized discrete values, and one must use a stacked column chart because the y variable is a percentage, grouped by a third variable (license status).

Don't be mad, just ditch the flowchart.

 

Deliberately obstructing chart elements as a plot point

These "ridge plots" have become quite popular in recent times. The following example, from this BBC report (link), shows the change in global air temperatures over time.

***

This chart is in reality a panel of probability density plots, one for each year of the dataset. The years are arranged with the oldest at the top and the most recent at the bottom. You take those plots and squeeze every ounce of the space out, so that each chart overlaps massively with the ones above it.

The plot at the bottom is the only one that can be seen unobstructed.

Overplotting chart elements, deliberately obstructing them, doesn't sound useful. Is there something gained for what's lost?

***

The appeal of the ridge plot is the metaphor of ridges, or crests if you see ocean waves. What do these features signify?

The legend at the bottom of the chart gives a hint.

The main metric used to describe global warming is the amount of excess temperature, defined as the temperature relative to a historical average, set as the average temperature during the pre-industrial age. In recent years, the average global temperature is about 1.5 degrees Celsius above the reference level.

One might think that the higher the peak in a given plot, the higher the excess temperature. Not so. The heights of those peaks do not indicate temperatures.

What's the scale of the vertical axis? The labels suggest years, but that's a distractor also. If we consider the panel of non-overlapping probability density charts, the vertical axis should show probability density. In such a panel, the year labels should go to the titles of individual plots. On the ridge plot, the density axes are sacrificed, while the year labels are shifted to the vertical axis.

Admittedly, probability density is not an intuitive concept, so not much is lost by its omission.

The legend appears to suggest that the vertical scale is expressed in number of days so that in any given year, the peak of the curve occurs where the most likely excess temperature is found. But the amount of excess is read from the horizontal axis, not the vertical axis - it is encoded as a displacement in location horizontally away from the historical average. In other words, the height of the peak still doesn't correlate with the magnitude of the excess temperature.

The following set of probability density curves (with made-up data) each has the same average excess temperature of 1.5 degrees. Going from top to bottom, the variability of the excess temperatures increases. The height of the peak decreases accordingly because in a density plot, we require the total area under the curve to be fixed. Thus, the higher the peak, the lower the daily variability of the excess temperature.

A problem with this ridge plot is that it draws our attention to the heights of the peaks, which provide information about a secondary metric.

If we want to find the story that the amount of excess temperature has been increasing over time, we would have to trace a curve through the ridges, which strangely enough is a line that moves top to bottom, initially somewhat vertically, then moving sideways to the right. In a more conventional chart, the line that shows growth over time moves from bottom left to top right.

***

The BBC article (link) features several charts. The first one shows how the average excess temperature trends year to year. This is a simple column chart. By supplementing the column chart with the ridge plot, I assume that the designer wants to tell readers that the average annual excess temperature masks daily variability. Therefore, each annual average has been disaggregated into 366 daily averages.

In the column chart, the annual average is compared to the historical average of 50 years. In the ridge plot, the daily average is compared to ... the same historical average of 50 years. That's what the reference line labeled pre-industrial average is saying to me.

It makes more sense to compare the 366 daily averages to 366 daily averages from those 50 years.

But now I've ruined the dataviz because in each probability density plot, there are 366 different reference points. But not really. We just have to think a little more abstractly. These 366 different temperatures are all mapped to the number zero, after adjustment. Thus, they all coincide at the same location on the horizontal axis.

(It's possible that they actually used 366 daily averages as references to construct the ridge plot. I'm guessing not but feel free to comment if you know how these values are computed.)

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.

 

Simple presentations

In the previous post, I looked at this chart that shows the distributions of four subgroups found in a dataset:

This chart takes quite some effort to decipher, as does another version I featured.

The key messages appear to be: (i) most English words are of Germanic origin, (ii) the most popular English words are even more skewed towards Germanic origin, (iii) words of French origin started showing up around rank 50, those of Latin origin around rank 250.

***

If we are making a graphic for presentation, we can simplify the visual clutter tremendously by - hmmm - a set of pie charts.

For those allergic to pies, here's a stacked column chart:

Both of these can be thought of as "samples" from the original chart, selected to highlight shifts in the relative proportions.

I also reversed the direction of the horizontal axis as I think the story is better told starting from the whole dataset and honing in on subsets.

 

P.S. [1/10/2025] A reader who has expertise in this subject also suggested a stacked column chart with reversed axis in a comment, so my recommendation here is confirmed.

Five-value summaries of distributions

BG commented on my previous post, describing her frustration with the “stacked range chart”:

A stacked graph visualizes cubes stacked one on top of the other. So you can't use it for negative numbers, because there's no such thing [as] "negative data". In graphs, a "minus" sign visualizes the opposite direction of one series from another. Doing average plus average plus average plus average doesn't seem logical at all.

***

I have already planned a second post to discuss the problems of using a stacked column chart to show markers of a numeric distribution.

I tried to replicate how the Youtuber generated his “stacked range chart” by appropriating Excel’s stacked column chart, but failed. I think there are some missing steps not mentioned in the video. At around 3:33 of the video, he shows a “hack” involving adding 100 degrees (any large enough value) to all values (already converted to ranges). Then, the next screen displays the resulting chart. Here is the dataset on the left and the chart on the right.

Afterwards, he replaces the axis labels with new labels, effectively shifting the axis. But something is missing from the narrative. Since he’s using a stacked column chart, the values in the table are encoded in the heights of the respective blocks. The total stacked heights of each column should be in the hundreds since he has added 100 to each cell. But that’s not what the chart shows.

***

In the rest of the post, I’ll skip over how to make such a chart in Excel, and talk about the consequences of inserting “range” values into the heights of the blocks of a stacked column chart.

Let’s focus on London, Ontario; the five temperature values, corresponding to various average temperatures, are -3, 5, 9, 14, 24. Just throwing those numbers into a stacked column chart in Excel results in the following useless chart:

The temperature averages are cumulatively summed, which makes no sense, as noted by reader BG. [My daily temperature data differ somewhat from those in the Youtube. My source is here.]

We should ignore the interiors of the blocks, and instead interpret the edges of these blocks. There are five edges corresponding to the five data values. As in:

The average temperature in London, Ontario (during Spring 2023-Winter 2024) is 9 C. This overall average hides seasonal as well as diurnal variations in temperature.

If we want to acknowledge that night-time temperatures are lower than day-time temperatures, we draw attention to the two values bracketing 9 C, i.e. 5 C and 14 C. The average daytime (max) temperature is 14 C while the average night-time (min) temperature is 5 C. Furthermore, Ontario experiences seasons, so that the average daytime temperature of 14 C is subject to seasonal variability; in the summer, it goes up to 24 C. In the winter, the average night-time temperature goes down to -3 C, compared to 5 C across all seasons. [For those paying closer attention, daytime/max and night-time/min form congruous pairs because the max temperature occurs during daytime while the min temperature occurs during night-time. Thus, the average of maximum temperatures is the same as the average of daytime maximum temperatures.]

The above dotplot illustrates this dataset adequately. The Youtuber explained why he didn’t like it – I couldn’t quite make sense of what he said. It’s possible he thinks the gaps between those averages are more meaningful than the averages themselves, and therefore he prefers a chart form that draws our attention to the ranges, rather than the values.

***

Our basic model of temperature can be thought of as: temperature on a given day = overall average + adjustment for seasonality + adjustment for diurnality.

Take the top three values 9, 14, 24 from above list. Starting at the overall average of 9 C, the analyst gets to 14 if he hones in on max daily temperatures, and to 24 if he further restricts the analysis to summer months (which have the higher temperatures). The second gap is 10 C, twice as large as the first gap of 5 C. Thus, the seasonal fluctuations have larger magnitude than daily fluctuations. Said differently, the effect of seasons on temperature is bigger than that of hour of day.

In interpreting the “ranges” or gaps between averages, narrow ranges suggest low variability while wider ranges suggest higher variability.

Here's a set of boxplots for the same data:

The boxplot "edges" also demarcate five values; they are not the same five values as defined by the Youtuber but both sets of five values describe the underlying distribution of temperatures.

 

P.S. For a different example of something similar, see this old post.

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.

 

Small tweaks that make big differences

It's one of those days that a web search led me to an unfamiliar corner, and I found myself poring over a pile of column charts that look like this:

This pair of charts appears to be canonical in a type of genetics analysis. I'll focus on the column chart up top.

The chart plots a variety of gene functions along the horizontal axis. These functions are classified into three broad categories, indicated using axis annotation.

What are some small tweaks that readers will enjoy?

***

First, use colors. Here is an example in which the designer uses color to indicate the function classes:

The primary design difference between these two column charts is using three colors to indicate the three function classes. This little change makes it much easier to recognize the ending of one class and the start of the other.

Color doesn't have to be limited to column areas. The following example extends the colors to the axis labels:

Again, just a smallest of changes but it makes a big difference.

***

It bugs me a lot that the long axis labels are printed in a slanted way, forcing every serious reader to read with slanted heads.

Slanting it the other way doesn't help:

Vertical labels are best read...

These vertical labels are best read while doing side planks.

***

I'm surprised the horizontal alignment is rather rare. Here's one:

 

Adjust, and adjust some more

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.

This is one material adjustment!

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.

 

 

 

 

Chart without an axis

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.