Reading this chart won't take as long as withdrawing troops from Afghanistan

Art sent me the following Economist chart, noting how hard it is to understand. I took a look, and agreed. It's an example of a visual representation that takes more time to comprehend than the underlying data.

Econ_theendisnear

The chart presents responses to 3 questions on a survey. For each question, the choices are Approve, Disapprove, and "Neither" (just picking a word since I haven't seen the actual survey question). The overall approval/disapproval rates are presented, and then broken into two subgroups (Democrats and Republicans).

The first hurdle is reading the scale. Because the section from 75% to 100% has been removed, we are left with labels 0, 25, 50, 75, which do not say percentages unless we've consumed the title and subtitle. The Economist style guide places the units of data in the subtitle instead of on
the axis itself.

Our attention is drawn to the thick lines, which represent the differences between approval and disapproval rates. These differences are signed: it matters whether the proportion approving is higher or lower than the proportion disapproving. This means the data are encoded in the order of the dots plus the length of the line segment between them.

The two bottom rows of the Afghanistan question demonstrates this mental challenge. Our brains have to process the following visual cues:

1) the two lines are about the same lengths

2) the Republican dots are shifted to the right by a little

3) the colors of the dots are flipped

What do they all mean?

Econ_theendofforever_subset

A chart runs in trouble when you need a paragraph to explain how to read it.

It's sometimes alright to make complicated data visualization that illustrates complicated concepts. What justifies it is the payoff. I wrote about the concept of return on effort in data visualization here.

The payoff for this chart escaped me. Take the Democratic response to troop withdrawal. About 3/4 of Democrats approve while 15% disapprove. The thick line says 60% more Democrats approve than disapprove.

***

Here, I show the full axis, and add a 50% reference line

Junkcharts_redo_econ_theendofforever_1

Small edits but they help visualize "half of", "three quarters of".

***

Next, I switch to the more conventional stacked bars.

Junkcharts_redo_econ_theendofforever_stackedbars

This format reveals some of the hidden data on the chart - the proportion answering neither approve/disapprove, and neither yes/no.

On the stacked bars visual, the proportions are counted from both ends while in the dot plot above, the proportions are measured from the left end only.

***

Read all my posts about Economist charts here

 


Finding the hidden information behind nice-looking charts

This chart from Business Insider caught my attention recently. (link)

Bi_householdwealthchart

There are various things they did which I like. The use of color to draw a distinction between the top 3 lines and the line at the bottom - which tells the story that the bottom 50% has been left far behind. Lines being labelled directly is another nice touch. I usually like legends that sit atop the chart; in this case, I'd have just written the income groups into the line labels.

Take a closer look at the legend text, and you'd notice they struggled with describing the income percentiles.

Bi_householdwealth_legend

This is a common problem with this type of data. The top and bottom categories are easy, as it's most natural to say "top x%" and "bottom y%". By doing so, we establish two scales, one running from the top, and the other counting from the bottom - and it's a head scratcher which scale to use for the middle categories.

The designer decided to lose the "top" and "bottom" descriptors, and went with "50-90%" and "90-99%". Effectively, these follow the "bottom" scale. "50-90%" is the bottom 50 to 90 percent, which corresponds to the top 10 to 50 percent. "90-99%" is the bottom 90-99%, which corresponds to the top 1 to 10%. On this chart, since we're lumping the top three income groups, I'd go with "top 1-10%" and "top 10-50%".

***

The Business Insider chart is easy to mis-read. It appears that the second group from the top is the most well-off, and the wealth of the top group is almost 20 times that of the bottom group. Both of those statements are false. What's confusing us is that each line represents very different numbers of people. The yellow line is 50% of the population while the "top 1%" line is 1% of the population. To see what's really going on, I look at a chart showing per-capita wealth. (Just divide the data of the yellow line by 50, etc.)

Redo_bihouseholdwealth_legend

For this chart, I switched to a relative scale, using the per-capita wealth of the Bottom 50% as the reference level (100). Also, I applied a 4-period moving average to smooth the line. The data actually show that the top 1% holds much more wealth per capita than all other income segments. Around 2011, the gap between the top 1% and the rest was at its widest - the average person in the top 1% is about 3,000 times wealthier than someone in the bottom 50%.

This chart raises another question. What caused the sharp rise in the late 2000s and the subsequent decline? By 2020, the gap between the top and bottom groups is still double the size of the gap from 20 years ago. We'd need additional analyses and charts to answer this question.

***

If you are familiar with our Trifecta Checkup, the Business Insider chart is a Type D chart. The problem with it is in how the data was analyzed.


A note to science journal editors: require better visuals

In reviewing a new small-scale study of the Moderna vaccine, I found this chart:

Modernahalfdoses_fig3a

This style of charts is quite common in scientific papers. And they are horrible. It irks me to think that some authors are forced to adopt such styles.

The study's main goal is to compare two half doses to two full doses of the Moderna vaccine. (To understand the science, read the post on my book blog.) The participants were stratified by age group. The vaccine is expected to work better for younger people than for older people. The point of the study isn't to measure the difference by age group, and so the age-group dimension is secondary.

Upon recognizing that, I reduce the number of colors from 4 to 2:

Junkcharts_redo_modernahalfdoses_1

Halving the number of colors presents no additional difficulty. The reader spends less time cross-referencing.

The existence of the Pbo (placebo) and Conv (convalescent plasma) columns on the sides is both unsightly and suboptimal. The "Conv" serves as a reference level for the amount of antibodies the vaccine stimulates in people. A better way to display reference levels is using reference lines.

Junkcharts_redo_modernahalfdoses_2color

The biggest problem with the chart is the log scale on the vertical axis. This isn't even a log-10 but a log-2. (Each tick is a doubling of value.)

Take the first set of columns as an example. The second column is clearly less than twice the height of the first column, and yet 25 is 3.5 times bigger than 7.  The third column is also visually less than double the size of the second column, and yet 189 is 7.5 times bigger than 25. The areas (heights) of the columns do not convey the right information about relative sizes of the underlying data.

Here's an amusing observation. The brown area shaded below is half of the entire area of the chart - if we reverted it to a linear scale. And yet there is not a single data point above 250 in the data so the brown area is entirely empty.

Junkcharts_redo_modernahalfdoses_logscale

An effect of a log scale is to compress the larger values of a dataset. That's what you're seeing here.

I now revisualize using dotplots:

Junkcharts_redo_modernahalfdoses_dotplotlinear

The version on the left retains the log scale while the right one (pun intended) reverts to the linear scale.

The biggest effect by far is the spike of antibodies between day 29 and 43 - which is after the second shot is administered. (For Moderna, the second shot is targeted for day 28.) In fact, it is during that window that the level of antibodies went from below the "conv" level (i.e. from natural infection) to far above.

The log-scale version buries this finding because it squeezes the large numbers on the chart. In addition, it artificially pulls the small numbers toward the "Conv" level. On the right chart, the second dot for 18-54, full doses is only at half the level of "Conv"  but it looks tantalizing close to the "Conv" level on the left chart.

The authors of the study also claim that there is negligible dropoff by 30 days after the second dose, i.e. between the third and fourth dots in each set. That may be so on the log-scale chart but on the linear chart, we see a moderate reduction. I don't believe the size of this study allows us to make a stronger conclusion but the claim of no dropoff is dubious.

The left chart also obscures the age-group differences. It appears as if all four sets show roughly the same pattern. With the linear scale, we notice that the vaccine clearly works better for the younger subgroup. As I discussed on the book blog, no one actually knows what level of antibodies constitutes "protection," and so I can't say whether that age-group difference has practical significance.

***

I recommend using log scales sparingly and carefully. They are a source of much mischief and misadventure.

 

 

 


Reading an infographic about our climate crisis

Let's explore an infographic by SCMP, which draws attention to the alarming temperature recorded at Verkhoyansk in Russia on June 20, 2020. The original work was on the back page of the printed newspaper, referred to in this tweet.

This view of the globe brings out the two key pieces of evidence presented in the infographic: the rise in temperature in unexpected places, and the shrinkage of the Arctic ice.

Scmp_russianheat_1a

A notable design decision is to omit the color scale. On inspection, the scale is present - it was sewn into the graphic.

Scmp_russianheat_colorscale

I applaud this decision as it does not take the reader's eyes away from the graphic. Some information is lost as the scale isn't presented in full details but I doubt many readers need those details.

A key takeaway is that the temperature in Verkhoyansk, which is on the edge of the Arctic Circle, was the same as in New Delhi in India on that day. We can see how the red was encroaching upon the Arctic Circle.

***Scmp_russianheat_2a

Next, the rapid shrinkage of the Arctic ice is presented in two ways. First, a series of maps.

The annotations are pared to the minimum. The presentation is simple enough such that we can visually judge that the amount of ice cover has roughly halved from 1980 to 2009.

A numerical measure of the drop is provided on the side.

Then, a line chart reinforces this message.

The line chart emphasizes change over time while the series of maps reveals change over space.

Scmp_russianheat_3a

This chart suggests that the year 2020 may break the record for the smallest ice cover since 1980. The maps of Australia and India provide context to interpret the size of the Arctic ice cover.

I'd suggest reversing the pink and black colors so as to refer back to the blue and pink lines in the globe above.

***

The final chart shows the average temperature worldwide and in the Arctic, relative to a reference period (1981-2000).

Scmp_russianheat_4

This one is tough. It looks like an area chart but it should be read as a line chart. The darker line is the anomaly of Arctic average temperature while the lighter line is the anomaly of the global average temperature. The two series are synced except for a brief period around 1940. Since 2000, the temperatures have been dramatically rising above that of the reference period.

If this is a stacked area chart, then we'd interpret the two data series as summable, with the sum of the data series signifying something interesting. For example, the market shares of different web browsers sum to the total size of the market.

But the chart above should not be read as a stacked area chart because the outside envelope isn't the sum of the two anomalies. The problem is revealed if we try to articulate what the color shades mean.

Scmp_russianheat_4_inset

On the far right, it seems like the dark shade is paired with the lighter line and represents global positive anomalies while the lighter shade shows Arctic's anomalies in excess of global. This interpretation only works if the Arctic line always sits above the global line. This pattern is broken in the late 1990s.

Around 1999, the Arctic's anomaly is negative while the global anomaly is positive. Here, the global anomaly gets the lighter shade while the Arctic one is blue.

One possible fix is to encode the size of the anomaly into the color of the line. The further away from zero, the darker the red/blue color.

 

 


These are the top posts of 2020

It's always very interesting as a writer to look back at a year's of posts and find out which ones were most popular with my readers.

Here are the top posts on Junk Charts from 2020:

How to read this chart about coronavirus risk

This post about a New York Times scatter plot dates from February, a time when many Americans were debating whether Covid-19 was just the flu.

Proportions and rates: we are no dupes

This post about a ArsTechnica chart on the effects of Covid-19 by age is an example of designing the visual to reflect the structure of the data.

When the pie chart is more complex than the data

This post shows a 3D pie chart which is worse than a 2D pie chart.

Twitter people upset with that Covid symptoms diagram

This post discusses some complicated graphics designed to illustrate complicated datasets on Covid-19 symptoms.

Cornell must remove the logs before it reopens in the fall

This post is another warning to think twice before you use log scales.

What is the price of objectivity?

This post turns an "objective" data visualization into a piece of visual story-telling.

The snake pit chart is the best election graphic ever

This post introduces my favorite U.S. presidential election graphic, designed by the FiveThirtyEight team.

***

Here is a list of posts that deserve more attention:

Locating the political center

An example of bringing readers as close to the insights as possible

Visualizing change over time

An example of designing data visualization to reflect the structure of multivariate data

Bloomberg made me digest these graphics slowly

An example of simple and thoughtful graphics

The hidden bad assumption behind most dual-axis time-series charts

Read this before you make a dual-axis chart

Pie chart conventions

Read this before you make a pie chart

***
Looking forward to bring you more content in 2021!

Happy new year.


Is this an example of good or bad dataviz?

This chart is giving me feelings:

Trump_mcconnell_chart

I first saw it on TV and then a reader submitted it.

Let's apply a Trifecta Checkup to the chart.

Starting at the Q corner, I can say the question it's addressing is clear and relevant. It's the relationship between Trump and McConnell's re-election. The designer's intended message comes through strongly - the chart offers evidence that McConnell owes his re-election to Trump.

Visually, the graphic has elements of great story-telling. It presents a simple (others might say, simplistic) view of the data - just the poll results of McConnell vs McGrath at various times, and the election result. It then flags key events, drawing the reader's attention to those. These events are selected based on key points on the timeline.

The chart includes wise design choices, such as no gridlines, infusing the legend into the chart title, no decimals (except for last pair of numbers, the intention of which I'm not getting), and leading with the key message.

I can nitpick a few things. Get rid of the vertical axis. Also, expand the scale so that the difference between 51%-40% and 58%-38% becomes more apparent. Space the time points in proportion to the dates. The box at the bottom is a confusing afterthought that reduces rather than assists the messaging.

But the designer got the key things right. The above suggestions do not alter the reader's expereince that much. It's a nice piece of visual story-telling, and from what I can see, has made a strong impact with the audience it is intended to influence.

_trifectacheckup_junkchartsThis chart is proof why the Trifecta Checkup has three corners, plus linkages between them. If we just evaluate what the visual is conveying, this chart is clearly above average.

***

In the D corner, we ask: what the Data are saying?

This is where the chart runs into several problems. Let's focus on the last two sets of numbers: 51%-40% and 58%-38%. Just add those numbers and do you notice something?

The last poll sums to 91%. This means that up to 10% of the likely voters responded "not sure" or some other candidate. If these "shy" voters show up at the polls as predicted by the pollsters, and if they voted just like the not shy voters, then the election result would have been 56%-44%, not 51%-40%. So, the 58%-38% result is within the margin of error of these polls. (If the "shy" voters break for McConnell in a 75%-25% split, then he gets 58% of the total votes.)

So, the data behind the line chart aren't suggesting that the election outcome is anomalous. This presents a problem with the Q-D and D-V green arrows as these pairs are not in sync.

***

In the D corner, we should consider the totality of the data available to the designer, not just what the designer chooses to utilize. The pivot of the chart is the flag annotating the "Trump robocall."

Here are some questions I'd ask the designer:

What else happened on October 31 in Kentucky?

What else happened on October 31, elsewhere in the country?

Was Trump featured in any other robocalls during the period portrayed?

How many robocalls were made by the campaign, and what other celebrities were featured?

Did any other campaign event or effort happen between the Trump robocall and election day?

Is there evidence that nothing else that happened after the robocall produced any value?

The chart commits the XYopia (i.e. X-Y myopia) fallacy of causal analysis. When the data analyst presents one cause and one effect, we are cued to think the cause explains the effect but in every scenario that is not a designed experiment, there are multiple causes at play. Sometimes, the more influential cause isn't the one shown in the chart.

***

Finally, let's draw out the connection between the last set of poll numbers and the election results. This shows why causal inference in observational data is such a beast.

Poll numbers are about a small number of people (500-1,000 in the case of Kentucky polls) who respond to polling. Election results are based on voters (> 2 million). An assumption made by the designer is that these polls are properly conducted, and their results are credible.

The chart above makes the claim that Trump's robocall gave McConnell 7% more votes than expected. This implies the robocall influenced at least 140,000 voters. Each such voter must fit the following criteria:

  • Was targeted by the Trump robocall
  • Was reached by the Trump robocall (phone was on, etc.)
  • Responded to the Trump robocall, by either picking up the phone or listening to the voice recording or dialing a call-back number
  • Did not previously intend to vote for McConnell
  • If reached by a pollster, would refuse to respond, or say not sure, or voting for McGrath or a third candidate
  • Had no other reason to change his/her behavior

Just take the first bullet for example. If we found a voter who switched to McConnell after October 31, and if this person was not on the robocall list, then this voter contributes to the unexpected gain in McConnell votes but weakens the case that the robocall influenced the election.

As analysts, our job is to find data to investigate all of the above. Some of these are easier to investigate. The campaign knows, for example, how many people were on the target list, and how many listened to the voice recording.

 

 

 

 


Bloomberg made me digest these graphics slowly

Ask the experts to name the success metric of good data visualization, and you will receive a dozen answers. The field doesn't have an all-encompassing metric. A useful reference is Andrew Gelman and Antony Urwin (2012) in which they discussed the tradeoff between beautiful and informative, which derives from the familiar tension between art and science.

For a while now, I've been intrigued by metrics that measure "effort". Some years ago, I described the concept of a "return on effort" in this post. Such a metric can be constructed like the dominating financial metric of return on investment. The investment here is an investment of time, of attention. I strongly believe that if the consumer judges a data visualization to be compelling, engaging or  ell constructed, s/he will expend energy to devour it.

Imagine grub you discard after the first bite, compared to the delicious food experienced slowly, savoring every last bit.

Bloomberg_ambridge_smI'm writing this post while enjoying the September issue of Bloomberg Businessweek, which focuses on the upcoming U.S. Presidential election. There are various graphics infused into the pages of the magazine. Many of these graphics operate at a level of complexity above what typically show up in magazines, and yet I spent energy learning to understand them. This response, I believe, is what visual designers should aim for.

***

Today, I discuss one example of these graphics, shown on the right. You might be shocked by the throwback style of these graphics. They look like they arrived from decades ago!

Grayscale, simple forms, typewriter font, all caps. Have I gone crazy?

The article argues that a town like Ambridge in Beaver County, Pennslyvania may be pivotal in the November election. The set of graphics provides relevant data to understand this argument.

It's evidence that data visualization does not need whiz-bang modern wizardry to excel.

Let me focus on the boxy charts from the top of the column. These:

Bloomberg_ambridge_topboxes

These charts solve a headache with voting margin data in the U.S.  We have two dominant political parties so in any given election, the vote share data split into three buckets: Democratic, Republican, and a catch-all category that includes third parties, write-ins, and none of the above. The third category rarely exceeds 5 percent.  A generic pie chart representation looks like this:

Redo_junkcharts_bloombergambridgebox_pies

Stacked bars have this look:

Redo_junkcharts_bloombergambridgebox_bars

In using my Trifecta framework (link), the top point is articulating the question. The primary issue here is the voting margin between the winner and the second-runner-up, which is the loser in what is typically a two-horse race. There exist two sub-questions: the vote-share difference between the top two finishers, and the share of vote effectively removed from the pot by the remaining candidates.

Now, take another look at the unusual chart form used by Bloomberg:

Bloomberg_ambridge_topboxes1

The catch-all vote share sits at the bottom while the two major parties split up the top section. This design demonstrates a keen understanding of the context. Consider the typical outcome, in which the top two finishers are from the two major parties. When answering the first sub-question, we can choose the raw vote shares, or the normalized vote shares. Normalizing shifts the base from all candidates to the top two candidates.

The Bloomberg chart addresses both scales. The normalized vote shares can be read directly by focusing only on the top section. In an even two-horse race, the top section is split by half - this holds true regardless of the size of the bottom section.

This is a simple chart that packs a punch.

 


Deaths as percent neither of cases nor of population. Deaths as percent of normal.

Yesterday, I posted a note about excess deaths on the book blog (link). The post was inspired by a nice data visualization by the New York Times (link). This is a great example of data journalism.

Nyt_excessdeaths_south

Excess deaths is a superior metric for measuring the effect of Covid-19 on public health. It's better than deaths as percent of cases. Also better than percent of the population.What excess deaths measure is deaths as a percent of normal. Normal is usually defined as the average deaths in the respective week in years past.

The red areas indicate how far the deaths in the Southern states are above normal. The highest peak, registered in Texas in late July, is 60 percent above the normal level.

***

The best way to appreciate the effort that went into this graphic is to imagine receiving the outputs from the model that computes excess deaths. A three-column spreadsheet with columns "state", "week number" and "estimated excess deaths".

The first issue is unequal population sizes. More populous states of course have higher death tolls. Transforming death tolls to an index pegged to the normal level solves this problem. To produce this index, we divide actual deaths by the normal level of deaths. So the spreadsheet must be augmented by two additional columns, showing the historical average deaths and actual deaths for each state for each week. Then, the excess death index can be computed.

The journalist builds a story around the migration of the coronavirus between different regions as it rages across different states  during different weeks. To this end, the designer first divides the dataset into four regions (South, West, Midwest and Northeast). Within each region, the states must be ordered. For each state, the week of peak excess deaths is identified, and the peak index is used to sort the states.

The graphic utilizes a small-multiples framework. Time occupies the horizontal axis, by convention. The vertical axis is compressed so that the states are not too distant. For the same reason, the component graphs are allowed to overlap vertically. The benefit of the tight arrangement is clearer for the Northeast as those peaks are particularly tall. The space-saving appearance reminds me of sparklines, championed by Ed Tufte.

There is one small tricky problem. In most of June, Texas suffered at least 50 percent more deaths than normal. The severity of this excess death toll is shortchanged by the low vertical height of each component graph. What forced such congestion is probably the data from the Northeast. For example, New York City:

Nyt_excessdeaths_northeast3

 

New York City's death toll was almost 8 times the normal level at the start of the epidemic in the U.S. If the same vertical scale is maintained across the four regions, then the Northeastern states dwarf all else.

***

One key takeaway from the graphic for the Southern states is the persistence of the red areas. In each state, for almost every week of the entire pandemic period, actual deaths have exceeded the normal level. This is strong indication that the coronavirus is not under control.

In fact, I'd like to see a second set of plots showing the cumulative excess deaths since March. The weekly graphic is better for identifying the ebb and flow while the cumulative graphic takes measure of the total impact of Covid-19.

***

The above description leaves out a huge chunk of work related to computing excess deaths. I assumed the designer receives these estimates from a data scientist. See the related post in which I explain how excess deaths are estimated from statistical models.

 


Everything in Texas is big, but not this BIG

Long-time reader John forwarded the following chart via Twitter.

Covidtracking_texassquare

The chart shows the recent explosive growth in deaths due to Covid-19 in Texas. John flagged this graphic as yet another example in which the data are encoded to the lengths of the squares, not their areas.

Fixing this chart just requires fixing the length of one side of the square. I also flipped it to make a conventional column chart.

Redo_texasdeathsquares_process

The final product:

Redo_texasdeaths_columns

An important qualification lurks in the footnote; it is directly applied to the label of July.

How much visual distortion is created when data are encoded to the lengths and not the areas? The following chart shows what readers see, assuming they correctly perceive the areas of those squares. The value for March is held the same as above while the other months show the death counts implied by the relative areas of the squares.

Redo_texasdeaths_distortion

Owing to squaring, the smaller counts are artificially compressed while the big numbers are massively exaggerated.


On data volume, reliability, uncertainty and confidence bands

This chart from the Economist caught my eye because of the unusual use of color-coded hexagonal tiles.

Economist_lifequalitywealth1

The basic design of the chart is easy to grasp: It relates people's "happiness" to national wealth. The thick black line shows that the average citizen of wealthier countries tends to rate their current life situation better.

For readers alert to graphical details, things can get a little confusing. The horizontal "wealth" axis is shown in log scale, which means that the data on the right side of the chart have been compressed while the data on the left side of the chart have been stretched out. In other words, the curve in linear scale is much flatter than depicted.

Redo_economistlifesatisfaction_linear

One thing you might notice is how poor the fit of the line is at both ends. Singapore and Afghanistan are clearly not explained by the fitted line. (That said, the line is based on many more dots than those eight we can see.) Moreover, because countries are widely spread out on the high end of the wealth axis, the fit is not impressive. Log scales tend to give a false impression of the tightness of fit, as I explained before when discussing coronavirus case curves.

***

The hexagonal tiles replace the more typical dot scatter or contour shading. The raw data consist of results from polls conducted in different countries in different years. For each poll, the analyst computes the average life satisfaction score for that country in that year. From national statistics, the analyst pulls out that country's GDP per capita in that year. Thus, each data point is a dot on the canvass. A few data points are shown as black dots. Those are for eight highlighted countries for the year 2018.

The black line is fitted to the underlying dot scatter and summarizes the correlation between average wealth and average life satisfaction. Instead of showing the scatter, this Economist design aggregates nearby dots into hexagons. The deepest red hexagon, sandwiched between Finland and the US, contains about 60-70 dots, according to the color legend.

These details are tough to take in. It's not clear which dots have been collected into that hexagon: are they all Finland or the U.S. in various years, or do they include other countries? Each country is represented by multiple dots, one for each poll year. It's also not clear how much variation there exists within a country across years.

***

The hexagonal tiles presumably serve the same role as a dot scatter or contour shading. They convey the amount of data supporting the fitted curve along its trajectory. More data confers more reliability.

For this chart, the hexagonal tiles do not add any value. The deepest red regions are those closest to the black line so nothing is actually lost by showing just the line and not the tiles.

Redo_economistlifesatisfaction_nohex

Using the line chart obviates the need for readers to figure out the hexagons, the polls, the aggregation, and the inevitable unanswered questions.

***

An alternative concept is to show the "confidence band" or "error bar" around the black line. These bars display the uncertainty of the data. The wider the band, the less certain the analyst is of the estimate. Typically, the band expands near the edges where we have less data.

Here is conceptually what we should see (I don't have the underlying dataset so can't compute the confidence band precisely)

Redo_economistlifesatisfaction_confband

The confidence band picture is the mirror image of the hexagonal tiles. Where the poll density is high, the confidence band narrows, and where poll density is low, the band expands.

A simple way to interpret the confidence band is to find the country's wealth on the horizontal axis, and look at the range of life satisfaction rating for that value of wealth. Now pick any number between the range, and imagine that you've just conducted a survey and computed the average rating. That number you picked is a possible survey result, and thus a valid value. (For those who know some probability, you should pick a number not at random within the range but in accordance with a Bell curve, meaning picking a number closer to the fitted line with much higher probability than a number at either edge.)

Visualizing data involves a series of choices. For this dataset, one such choice is displaying data density or uncertainty or neither.