The one thing you're afraid to ask about histograms

In the previous post about a variant of the histogram, I glossed over a few perplexing issues - deliberately. Today's post addresses one of these topics: what is going on in the vertical axis of a histogram?

The real question is: what data are encoded in the histogram, and where?

***

Let's return to the dataset from the last post. I grabbed data from a set of international football (i.e. soccer) matches. Each goal scored has a scoring minute. If the goal is scored in regulation time, the scoring minute is a number between 1 and 90 minutes. Specifically, the data collector applies a rounding up: any goal scored between 0 and 60 seconds is recorded as 1, all the way up to a goal scored between 89 and 90th minute being recorded as 90. In this post, I only consider goals scored in regulation time so the horizontal axis is between 1-90 minutes.

The kneejerk answer to the posed question is: counts in bins. Isn't it the case that in constructing a histogram, we divide the range of values (1-90) into bins, and then plot the counts within bins, i.e. the number of goals scored within each bin of minutes?

The following is what we have in mind:

Let's call this the "count histogram".

Some readers may dislike the scale of the vertical axis, as its interpretation hinges on the total sample size. Hence, another kneejerk answer is: frequencies in bins. Instead of plotting counts directly, plot frequencies, which are just standardized counts. Just divide each value by the sample size. Here's the "frequency histogram":

The count and frequency histograms are identical except for the scale, and appear intuitively clear. The count and frequency data are encoded in the heights of the columns. The column widths are an afterthought, and they adhere to a fixed constant. Unlike a column chart, typically the gap width in a histogram is zero, as we want to partition the horizontal range into adjoining sections.

Now, if you look carefully at the histogram from the last post, reproduced below, you'd find that it plots neither counts nor frequencies:

The numbers on the axis are fractions, and suggest that they may be frequencies, but a quick check proves otherwise: with 9 columns, the average column should contain at least 10 percent of the data. The total of the displayed fractions is nowhere near 100%, which is our expectation if the values are relative frequencies. You may have come across this strangeness when creating histograms using R or some other software.

The purpose of this post is to explain what values are being plotted and why.

***

What are the kinds of questions we like to answer about the distribution of data?

At a high level, we want to know "where are my data"?

Arguably these two questions are fundamental:

• what is the probability that the data falls within a given range of values? e.g., what is the probability that a goal is scored in the first 15 minutes of a football match?
• what is the relative probability of data between two ranges of values? e.g. are teams more likely to score in last 5 minutes of the first half or the last five minutes of the second half of a football match?

In a histogram, the first question is answered by comparing a given column to the entire set of columns while the second question is answered by comparing one column to another column.

Let's see what we can learn from the count histogram.

In a count histogram, the heights encode the count data. To address the relative probability question, we note that the ratio of heights is the ratio of counts, and the ratio of counts is the same as the ratio of frequencies. Thus, we learn that teams are roughly 3000/1500 = 1.5 times more likely to score in the last 5 minutes of the second half than during the last 5 minutes of the first half. (See the green columns).

[For those who follow football, it's clear that the data collector treated goals scored during injury time of either half as scored during the last minute of the half, so this dataset can't be used to analyze timing of goals unless the real minutes were recorded for injury-time goals.]

To address the range probability question, we compare the aggregate height of the three orange columns with the total heights of all columns. Note that I said "height", not "area," because the heights directly encode counts. It's actually taxing to figure out the total height!

We resort to reading the total area of all columns. This should yield the correct answer: the area is directly proportional to the height because the column widths are fixed as a constant. Bear in mind, though, if the column widths vary (the theme of the last post), then areas and heights are not interchangable concepts.

Estimating the total area is still not easy, especially if the column heights exhibit high variance. What we need is the proportion of the total area that is orange. It's possible to see, not easy.

You may interject now to point out that the total area should equal the aggregate count (sample size). But that is a fallacy! It's very easy to make this error. The aggregate count is actually the total height, and because of that, the total area is the aggregate count multiplied by the column width! In my example, the total height is 23,682, which is the number of goals in the dataset, while the total area is 23,682 times 5 minutes.

[For those who think in equations, the total area is the sum over all columns of height(i) x width(i). When width is constant, we can take it outside the sum, and the sum of height(i) is just the total count.]

***

The count histogram is hard to use because it requires knowing the sample size. It's the first thing that is produced because the raw data are counts in bins. The frequency histogram is better at delivering answers.

In the frequency histogram, the heights encode frequency data. We can therefore just read off the relative probability of the orange column, bypassing the need to compute the total area.

This workaround actually promotes the fallacy described above for the count histogram. It is easy to fall into the trap of thinking that the total area of all columns is 100%. It isn't.

Similar to before, the total height should be the total frequency but the total area is the total frequency multipled by the column width, that is to say, the total area is the reciprocal of the bin width. In the football example, using 5-minute intervals, the total area of the frequency histogram is 1/(5 minutes) in the case of equal bin widths.

How about the relative probability question? On the frequency histogram, the ratio of column heights is the ratio of frequencies, which is exactly what we want. So long as the column width is constant, comparing column heights is easy.

***

One theme in the above discussion is that in the count and frequency histograms, the count and frequency data are encoded in the column heights but not the column areas. This is a source of major confusion. Because of the convention of using equal column widths, one treats areas and heights as interchangable... but not always. The total column area isn't the same as the total column height.

This observation has some unsettling implications.

As shown above, the total area is affected by the column width. The column width in an equal-width histogram is the range of the x-values divided by the number of bins. Thus, the total area is a function of the number of bins.

Consider the following frequency histograms of the same scoring minutes dataset. The only difference is the number of bins used.

Increasing the number of bins has a series of effects:

• the columns become narrower
• the columns become shorter, because each narrower bin can contain at most the same count as the wider bin that contains it.
• the total area of the columns become smaller.

This last one is unexpected and completely messes up our intuition. When we increase the number of bins, not only are the columns shortening but the total area covered by all the columns is also shrinking. Remember that the total area whether it is a count or frequency histogram has a factor equal to the bin width. Higher number of bins means smaller bin width, which means smaller total area.

***

What if we force the total area to be constant regardless of how many bins we use? This setting seems more intuitive: in the 5-bin histogram, we partition the total area into five parts while in the 10-bin histogram, we divide it into 10 parts.

This is the principle used by R and the other statistical software when they produce so-called density histograms. The count and frequency data are encoded in the column areas - by implication, the same data could not have been encoded simultaneously in the column heights!

The way to accomplish this is to divide by the bin width. If you look at the total area formulas above, for the count histogram, total area is total count x bin width. If the height is count divided by bin width, then the total area is the total count. Similarly, if the height in the frequency histogram is frequency divided by bin width, then the total area is 100%.

Count divided by some section of the x-range is otherwise known as "density". It captures the concept of how tightly the data are packed inside a particular section of the dataset. Thus, in a count-density histogram, the heights encode densities while the areas encode counts. In this case, total area is the total count. If we want to standardize total area to be 1, then we should compute densities using frequencies rather than counts. Frequency densities are just count densities divided by the total count.

To summarize, in a frequency-density histogram, the heights encode densities, defined as frequency divided by the bin width. This is not very intuitive; just think of densities as how closely packed the data are in the specified bin. The column areas encode frequencies so that the total area is 100%.

The reason why density histograms are confusing is that we are reading off column heights while thinking that the total area should add up to 100%. Column heights and column areas cannot both add up to 100%. We have to pick one or the other.

Comparing relative column heights still works when the density histogram has equal bin widths. In this case, the relative height and relative area are the same because relative density equals relative frequencies if the bin width is fixed.

The following charts recap the discussion above. It shows how the frequency histogram does not preserve the total area when bin sizes are changed while the density histogram does.

***

The density histogram is a major pain for solving range probability questions because the frequencies are encoded in the column areas, not the heights. Areas are not marked out in a graph.

The column height gives us densities which are not probabilities. In order to retrieve probabilities, we have to multiply the density by the bin width, that is to say, we must estimate the area of the column. That requires mapping two dimensions (width, height) onto one (area). It is in fact impossible without measurement - unless we make the bin widths constant.

When we make the bin widths constant, we still can't read densities off the vertical axis, and treat them as probabilities. If I must use the density histogram to answer the question of how likely a team scores in the first 15 minutes, I'd sum the heights of the first 3 columns, which is about 0.025, and then multiply it by the bin width of 5 minutes, which gives 0.125 or 12.5%.

At the end of this exploration, I like the frequency histogram best. The density histogram is useful when we are comparing different histograms, which isn't the most common use case.

***

The histogram is a basic chart in the tool kit. It's more complicated than it seems. I haven't come across any intro dataviz books that explain this clearly.

Most of this post deals with equal-width histograms. If we allow bin widths to vary, it gets even more complicated. Stay tuned.

***

For those using base R graphics, I hope this post helps you interpret what they say in the manual. The default behavior of the "hist" function depends on whether the bins are equal width:

• if the bin width is constant, then R produces a count histogram. As shown above, in a count histogram, the column heights indicate counts in bins but the total column area does not equal the total sample size, but the total sample size multiplied by the bin width. (Equal width is the default unless the user specifies bin breakpoints.)
• if the bin width is not constant, then R produces a (frequency-)density histogram. The column heights are densities, defined as frequencies divided by bin width while the column areas are frequencies, with the total area summing to 100%.

Unfortunately, R does not generate a frequency histogram. To make one, you'd have to divide the counts in bins by the sum of counts. (In making some of the graphs above, I tricked it.) You also need to trick it to make a frequency-density histogram with equal-width bins, as it's coded to produce a count histogram when bin size is fixed.

 

P.S. [5-2-2023] As pointed out by a reader, I should clarify that R and I use the word "frequency" differently. Specifically, R uses frequency to mean counts, therefore, what I have been calling the "count histogram", R would have called it a "frequency histogram", and what I have been describing as a "frequency histogram", the "hist" function simply does not generate it unless you trick it to do so. I'm using "frequency" in the everyday sense of the word, such as "the frequency of the bus". In many statistical packages, frequency is used to mean "count", as in the frequency table which is just a table of counts. The reader suggested proportion which I like, or something like weight.

 

 

 

 

 

Bivariate choropleths

A reader submitted a link to Joshua Stephen's post about bivariate choropleths, which is the technical term for the map that FiveThirtyEight printed on abortion bans, discussed here. Joshua advocates greater usage of maps with two-dimensional color scales.

As a reminder, the fundamental building block is expressed in this bivariate color legend:

Counties are classified into one of these nine groups, based on low/middle/high ratings on two dimensions, distance and congestion.

The nine groups are given nine colors, built from superimposing shades of green and pink. All nine colors are printed on the same map.

Without a doubt, using these nine related colors are better than nine arbitrary colors. But is this a good data visualization?

Specifically, is the above map better than the pair of maps below?

The split map is produced by Josh to explain that the bivariate choropleth is just the superposition of two univariate choropleths. I much prefer the split map to the superimposed one.

***

Think about what the reader goes through when comparing two counties.

Superimposing the two univariate maps solves one problem: it removes the need to scan back and forth between two maps, looking for the same locations, something that is imprecise. (Unless, the map is interactive, and highlighting one county highlights the same county in the other map.)

For me, that's a small price to pay for quicker translation of color into information.

 

 

Finding the story in complex datasets

In CT Mirror's feature about Connecticut, which I wrote about in the previous post, there is one graphic that did not rise to the same level as the others.

This section deals with graduation rates of the state's high school districts. The above chart focuses on exactly five districts. The line charts are organized in a stack. No year labels are provided. The time window is 11 years from 2010 to 2021. The column of numbers show the difference in graduation rates over the entire time window.

The five lines look basically the same, if we ignore what looks to be noisy year-to-year fluctuations. This is due to the weird aspect ratio imposed by stacking.

Why are those five districts chosen? Upon investigation, we learn that these are the five districts with the biggest improvement in graduation rates during the 11-year time window.

The same five schools also had some of the lowest graduation rates at the start of the analysis window (2010). This must be so because if a school graduated 90% of its class in 2010, it would be mathematically impossible for it to attain a 35% percent point improvement! This is a dissatisfactory feature of the dataviz.

***

In preparing an alternative version, I start by imagining how readers might want to utilize a visualization of this dataset. I assume that the readers may have certain school(s) they are particularly invested in, and want to see its/their graduation performance over these 11 years.

How does having the entire dataset help? For one thing, it provides context. What kind of context is relevant? As discussed above, it's futile to compare a school at the top of the ranking to one that is near the bottom. So I created groups of schools. Each school is compared to other schools that had comparable graduation rates at the start of the analysis period.

Amistad School District, which takes pole position in the original dataviz, graduated only 58% of its pupils in 2010 but vastly improved its graduation rate by 35% over the decade. In the chart below (left panel), I plotted all of the schools that had graduation rates between 50 and 74% in 2010. The chart shows that while Amistad is a standout, almost all schools in this group experienced steady improvements. (Whether this phenomenon represents true improvement, or just grade inflation, we can't tell from this dataset alone.)

The right panel shows the group of schools with the next higher level of graduation rates in 2010. This group of schools too increased their graduation rates almost always. The rate of improvement in this group is lower than in the previous group of schools.

The next set of charts show school districts that already achieved excellent graduation rates (over 85%) by 2010. The most interesting group of schools consists of those with 85-89% rates in 2010. Their performance in 2021 is the most unpredictable of all the school groups. The majority of districts did even better while others regressed.

Overall, there is less variability than I'd expect in the top two school groups. They generally appeared to have been able to raise or maintain their already-high graduation rates. (Note that the scale of each chart is different, and many of the lines in the second set of charts are moving within a few percentages.)

One more note about the charts: The trend lines are "smoothed" to focus on the trends rather than the year to year variability. Because of smoothing, there is some awkward-looking imprecision e.g. the end-to-end differences read from the curves versus the observed differences in the data. These discrepancies can easily be fixed if these charts were to be published.

Thoughts on Daniel's fix for dual-axes charts

I've taken a little time to ponder Daniel Z's proposed "fix" for dual-axes charts (link). The example he used is this:

In that long post, Daniel explained why he preferred to mix a line with columns, rather than using the more common dual lines construction: to prevent readers from falsely attributing meaning to crisscrossing lines. There are many issues with dual-axes charts, which I won't repeat in this post; one of their most dissatisfying features is the lack of connection between the two vertical scales, and thus, it's pretty easy to manufacture an image of correlation when it doesn't exist. As shown in this old post, one can expand or restrict one of the vertical axes and shift the line up and down to "match" the other vertical axis.

Daniel's proposed fix retains the dual axes, and he even restores the dual lines construction.

How is this chart different from the typical dual-axes chart, like the first graph in this post?

Recall that the problem with using two axes is that the designer could squeeze, expand or shift one of the axes in any number of ways to manufacture many realities. What Daniel effectively did here is selecting one specific way to transform the "New Customers" axis (shown in gray).

His idea is to run a simple linear regression between the two time series. Think of fitting a "trendline" in Excel between Revenues and New Customers. Then, use the resulting regression equation to compute an "estimated" revenues based on the New Customers series. The coefficients of this regression equation then determines the degree of squeezing/expansion and shifting applied to the New Customers axis.

The main advantage of this "fix" is to eliminate the freedom to manufacture multiple realities. There is exactly one way to transform the New Customers axis.

The chart itself takes a bit of time to get used to. The actual values plotted in the gray line are "estimated revenues" from the regression model, thus the blue axis values on the left apply to the gray line as well. The gray axis shows the respective customer values. Because we performed a linear fit, each value of estimated revenues correspond to a particular customer value. The gray line is thus a squeezed/expanded/shifted replica of the New Customers line (shown in orange in the first graph). The gray line can then be interpreted on two connected scales, and both the blue and gray labels are relevant.

***

What are we staring at?

The blue line shows the observed revenues while the gray line displays the estimated revenues (predicted by the regression line). Thus, the vertical gaps between the two lines are the "residuals" of the regression model, i.e. the estimation errors. If you have studied Statistics 101, you may remember that the residuals are the components that make up the R-squared, which measures the quality of fit of the regression model. R-squared is the square of r, which stands for the correlation between Customers and the observed revenues. Thus the higher the (linear) correlation between the two time series, the higher the R-squared, the better the regression fit, the smaller the gaps between the two lines.

***

There is some value to this chart, although it'd be challenging to explain to someone who has not taken Statistics 101.

While I like that this linear regression approach is "principled", I wonder why this transformation should be preferred to all others. I don't have an answer to this question yet.

***

Daniel's fix reminds me of a different, but very common, chart.

This chart shows actual vs forecasted inflation rates. This chart has two lines but only needs one axis since both lines represent inflation rates in the same range.

We can think of the "estimated revenues" line above as forecasted or expected revenues, based on the actual number of new customers. In particular, this forecast is based on a specific model: one that assumes that revenues is linearly related to the number of new customers. The "residuals" are forecasting errors.

In this sense, I think Daniel's solution amounts to rephrasing the question of the chart from "how closely are revenues and new customers correlated?" to "given the trend in new customers, are we over- or under-performing on revenues?"

Instead of using the dual-axes chart with two different scales, I'd prefer to answer the question by showing this expected vs actual revenues chart with one scale.

This does not eliminate the question about the "principle" behind the estimated revenues, but it makes clear that the challenge is to justify why revenues is a linear function of new customers, and no other variables.

Unlike the dual-axes chart, the actual vs forecasted chart is independent of the forecasting method. One can produce forecasted revenues based on a complicated function of new customers, existing customers, and any other factors. A different model just changes the shape of the forecasted revenues line. We still have two comparable lines on one scale.

 

 

 

 

 

All about Connecticut

This dataviz project by CT Mirror is excellent. The project walks through key statistics of the state of Connecticut.

Here are a few charts I enjoyed.

The first one shows the industries employing the most CT residents. The left and right arrows are perfect, much better than the usual dot plots.

The industries are sorted by decreasing size from top to bottom, based on employment in 2019. The chosen scale is absolute, showing the number of employees. The relative change is shown next to the arrow heads in percentages.

The inclusion of both absolute and relative scales may be a source of confusion as the lengths of the arrows encode the absolute differences, not the relative differences indicated by the data labels. This type of decision is always difficult for the designer. Selecting one of the two scales may improve clarity but induce loss aversion.

***

The next example is a bumps chart showing the growth in residents with at least a bachelor's degree.

This is more like a slopegraph as it appears to draw straight lines between two time points 9 years apart, omitting the intervening years. Each line represents a state. Connecticut's line is shown in red. The message is clear. Connecticut is among the most highly educated out of the 50 states. It maintained this advantage throughout the period.

I'd prefer to use solid lines for the background states, and the axis labels can be sparser.

It's a little odd that pretty much every line has the same slope. I'm suspecting that the numbers came out of a regression model, with varying slopes by state, but the inter-state variance is low.

In the online presentation, one can click on each line to see the values.

***

The final example is a two-sided bar chart:

This shows migration in and out of the state. The red bars represent the number of people who moved out, while the green bars represent those who moved into the state. The states are arranged from the most number of in-migrants to the least.

I have clipped the bottom of the chart as it extends to 50 states, and the bottom half is barely visible since the absolute numbers are so small.

I'd suggest showing the top 10 states. Then group the rest of the states by region, and plot them as regions. This change makes the chart more compact, as well as more useful.

***

There are many other charts, and I encourage you to visit and support this data journalism.

 

 

 

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)

 

 

Visual cues affect how data are perceived

Here's a recent NYT graphic showing California's water situation at different time scales (link to article).

It's a small multiples display, showing the spatial distribution of the precipitation amounts in California. The two panels show, respectively, the short-term view (past month) and the longer-term view (3 years). Precipitation is measured in relative terms,  so what is plotted is the relative ratio of precipitation in the reference period, with 100 being the 30-year average.

Green is much wetter than average while brown is much drier than average.

The key to making this chart work is a common color scheme across the two panels.

Also, the placement of major cities provides anchor points for our eyes to move back and forth between the two panels.

***

The NYT graphic is technically well executed. I'm a bit unhappy with the headline: "Recent rains haven't erased California's long-term drought".

At the surface, the conclusion seems sensible. Look, there is a lot of green, even deep green, on the left panel, which means the state got lots more rain than usual in the past month. Now, on the right panel, we find patches of brown, and very little green.

But pay attention to the scale. The light brown color, which covers the largest area, has value 70 to 90, thus, these regions have gotten 10-30% less precipitation than average in the past three years relative to the 30-year average.

Here's the question: what does it mean by "erasing California's long-term drought"? Does the 3-year average have to equal or exceed the 30-year average? Why should that be the case?

If we took all 3-year windows within those 30 years, we're definitely not going to find that each such 3-year average falls at or above the 30-year average. To illustrate this, I pulled annual rainfall data for San Francisco. Here is a histogram of 3-year averages for the 30-year period 1991-2020.

For example, the first value is the average rainfall for years 1989, 1990 and 1991, the next value is the average of 1990, 1991, and 1992, and so on. Each value is a relative value relative to the overall average in the 30-year window. There are two more values beyond 2020 that is not shown in the histogram. These are 57%, and 61%, so against the 30-year average, those two 3-year averages were drier than usual.

The above shows the underlying variability of the 3-year averages inside the reference time window. We have to first define "normal", and that might be a value between 70% and 130%.

In the same way, we can establish the "normal" range for the entire state of California. If it's also 70% to 130%, then the last 3 years as shown in the map above should be considered normal.

 

 

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

 

 

Painting the corner

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

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

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

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

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

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

***

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

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

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

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

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

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

 

 

 

Finding the right context to interpret household energy data

Bloomberg's recent article on surging UK household energy costs, projected over this winter, contains data about which I have long been intrigued: how much energy does different household items consume?

A twitter follower alerted me to this chart, and she found it informative.

***
If the goal is to pick out the appliances and estimate the cost of running them, the chart serves its purpose. Because the entire set of data is printed, a data table would have done equally well.

I learned that the mobile phone costs almost nothing to charge: 1 pence for six hours of charging, which is deemed a "single use" which seems double what a full charge requires. The games console costs 14 pence for a "single use" of two hours. That might be an underestimate of how much time gamers spend gaming each day.

***

Understanding the design of the chart needs a bit more effort. Each appliance is measured by two metrics: the number of hours considered to be "single use", and a currency value.

It took me a while to figure out how to interpret these currency values. Each cost is associated with a single use, and the duration of a single use increases as we move down the list of appliances. Since the designer assumes a fixed cost of electicity (shown in the footnote as 34p per kWh), at first, it seems like the costs should just increase from top to bottom. That's not the case, though.

Something else is driving these numbers behind the scene, namely, the intensity of energy use by appliance. The wifi router listed at the bottom is turned on 24 hours a day, and the daily cost of running it is just 6p. Meanwhile, running the fridge and freezer the whole day costs 41p. Thus, the fridge&freezer consumes electricity at a rate that is almost 7 times higher than the router.

The chart uses a split axis, which artificially reduces the gap between 8 hours and 24 hours. Here is another look at the bottom of the chart:

***

Let's examine the choice of "single use" as a common basis for comparing appliances. Consider this:

• Continuous appliances (wifi router, refrigerator, etc.) are denoted as 24 hours, so a daily time window is also implied
• Repeated-use appliances (e.g. coffee maker, kettle) may be run multiple times a day
• Infrequent use appliances may be used less than once a day

I prefer standardizing to a "per day" metric. If I use the microwave three times a day, the daily cost is 3 x 3p = 9 p, which is more than I'd spend on the wifi router, run 24 hours. On the other hand, I use the washing machine once a week, so the frequency is 1/7, and the effective daily cost is 1/7 x 36 p = 5p, notably lower than using the microwave.

The choice of metric has key implications on the appearance of the chart. The bubble size encodes the relative energy costs. The biggest bubbles are in the heating category, which is no surprise. The next largest bubbles are tumble dryer, dishwasher, and electric oven. These are generally not used every day so the "per day" calculation would push them lower in rank.

***

Another noteworthy feature of the Bloomberg chart is the split legend. The colors divide appliances into five groups based on usage category (e.g. cleaning, food, utility). Instead of the usual color legend printed on a corner or side of the chart, the designer spreads the category labels around the chart. Each label is shown the first time a specific usage category appears on the chart. There is a presumption that the reader scans from top to bottom, which is probably true on average.

I like this arrangement as it delivers information to the reader when it's needed.