Election visual 3: a strange, mash-up visualization

Continuing our review of FiveThirtyEight's election forecasting model visualization (link), I now look at their headline data visualization. (The previous posts in this series are here, and here.)

It's a set of 22 maps, each showing one election scenario, with one candidate winning. What chart form is this?

Small multiples may come to mind. A small-multiples chart is a grid in which every component graphic has the same form - same chart type, same color scheme, same scale, etc. The only variation from graphic to graphic is the data. The data are typically varied along a dimension of interest, for example, age groups, geographic regions, years. The following small-multiples chart, which I praised in the past (link), shows liquor consumption across the world.

Each component graphic changes according to the data specific to a country. When we scan across the grid, we draw conclusions about country-to-country variations. As with convention, there are as many graphics as there are countries in the dataset. Sometimes, the designer includes only countries that are directly relevant to the chart's topic.

***

What is the variable FiveThirtyEight chose to vary from map to map? It's the scenario used in the election forecasting model.

This choice is unconventional. The 22 scenarios is a subset of the 40,000 scenarios from the simulation - we are left wondering how those 22 are chosen.

Returning to our question: what chart form is this?

Perhaps you're reminded of the dot plot from the previous post. On that dot plot, the designer summarized the results of 40,000 scenarios using 100 dots. Since Biden is the winner in 75 percent of all scenarios, the dot plot shows 75 blue dots (and 25 red).

The map is the new dot. The 75 blue dots become 16 blue maps (rounded down) while the 25 red dots become 6 red maps.

Is it a pictogram of maps? If we ignore the details on the maps, and focus on the counts of colors, then yes. It's just a bit challenging because of the hole in the middle, and the atypical number of maps.

As with the dot plot, the map details are a nice touch. It connects readers with the simulation model which can feel very abstract.

Oddly, if you're someone familiar with probabilities, this presentation is quite confusing.

With 40,000 scenarios reduced to 22 maps, each map should represent 1818 scenarios. On the dot plot, each dot should represent 400 scenarios. This follows the rule for creating pictograms. Each object in a pictogram - dot, map, figurine, etc. - should encode an equal amount of the data. For the 538 visualization, is it true that each of the six red maps represents 1818 scenarios? This may be the case but not likely.

Recall the dot plot where the most extreme red dot shows a scenario in which Trump wins 376 out of 538 electoral votes (margin = 214). Each dot should represent 400 scenarios. The visualization implies that there are 400 scenarios similar to the one on display. For the grid of maps, the following red map from the top left corner should, in theory, represent 1,818 similar scenarios. Could be, but I'm not sure.

Mathematically, each of the depicted scenario, including the blowout win above, occurs with 1/40,000 chance in the simulation. However, one expects few scenarios that look like the extreme scenario, and ample scenarios that look like the median scenario.  

So, the right way to read the 538 chart is to ignore the map details when reading the embedded pictogram, and then look at the small multiples of detailed maps bearing in mind that extreme scenarios are unique while median scenarios have many lookalikes.

(Come to think about it, the analogous situation in the liquor consumption chart is the relative population size of different countries. When comparing country to country, we tend to forget that the data apply to large numbers of people in populous countries, and small numbers in tiny countries.)

***

There's a small improvement that can be made to the detailed maps. As I compare one map to the next, I'm trying to pick out which states that have changed to change the vote margin. Conceptually, the number of states painted red should decrease as the winning margin decreases, and the states that shift colors should be the toss-up states.

So I'd draw the solid Republican (Democratic) states with a lighter shade, forming an easily identifiable bloc on all maps, while the toss-up states are shown with a heavier shade.

Here, I just added a darker shade to the states that disappear from the first red map to the second.

Ask how you can give

A reader and colleague Georgette A was frustrated with the following graphic that appeared in the otherwise commendable article in National Geographic (link). The NatGeo article provides a history lesson on past pandemics that killed millions.

What does the design want to convey to readers?

Our attention is drawn to the larger objects, the red triangle on the left or the green triangle on the right. Regarding the red triangle, we learn that the base is the duration of the pandemic while the height of the black bar represents the total deaths.

An immediate curiosity is why a green triangle is lodged in the middle of the red triangle. Answering this question requires figuring out the horizontal layout. Where we expect axis labels we find an unexpected series of numbers (0, 16, 48, 5, 2, 4, ...). These are durations that measure the widths of the triangular bases.

To solve this puzzle, imagine the chart with the triangles removed, leaving just the black columns. Now replace the durations with index numbers, 1 to 13, corresponding to the time order of the ending years of these epidemics. In other words, there is a time axis hidden behind the chart. [As Ken reminded me on Twitter, I forgot to mention that details of each pandemic are revealed by hovering over each triangle.]

This explains why the green triangle (Antonine Plague) is sitting inside the large red triangle (Plague of Justinian). The latter's duration is 3 times that of the former, and the Antonine Plague ended before the Plague of Justinian. In fact, the Antonine occurred during 165-180 while the Justinian happened during 541-588. The overlap is an invention of the design. To receive what the design gives, we have to think of time as a sequence, not of dates.

***

Now, compare the first and second red triangles. Their black columns both encode 50 million deaths. The Justinian Plague however was spread out over 48 years while the Black Death lasted just 5 years. This suggests that the Black Death was more fearsome than the Justinian Plague. And yet, the graphic presents the opposite imagery.

This is a pretty tough dataset to visualize. Here is a side-by-side bar chart that lets readers first compare deaths, and then compare durations.

In the meantime, I highly recommend the NatGeo article.

Working with multiple dimensions, an example from Germany

An anonymous reader submitted this mirrored bar chart about violent acts by extremists in the 16 German states.

At first glance, this looks like a standard design. On a second look, you might notice what the reader discovered- the chart used two different scales, one for each side. The left side (red) depicting left-wing extremism is artificially compressed relative to the right side (blue). Not sure if this reflects the political bias of the publication - but in any case, this distortion means the only way to consume this chart is to read the numbers.

Even after fixing the scales, this design is challenging for the reader. It's unnatural to compare two years by looking first below then above. It's not simple to compare across states, and even harder to compare left- and right-wing extremism (due to mirroring).

The chart feels busy because the entire dataset is printed on it. I appreciate not including a redundant horizontal axis. (I wonder if the designer first removed the axis, then edited the scale on one side, not realizing the distortion.) Another nice touch, hidden in the legend, is the country totals.

I present two alternatives.

The first is a small-multiples "bumps chart".

Each plot presents the entire picture within a state. You can see the general level of violence, the level of left- and right-wing extremism, and their year-on-year change. States can be compared holistically.

Several German state names are rather long, so I explored a horizontal orientation. In this case, a connected dot plot may be more appropriate.

The sign of a good multi-dimensional visual display is whether readers can easily learn complex relationships. Depending on the question of interest, the reader can mentally elevate parts of this chart. One can compare the set of blue arrows to the set of red arrows, or focus on just blue arrows pointing right, or red arrows pointing left, or all arrows for Berlin, etc.

 

[P.S. Anonymous reader said the original chart came from the Augsburger newspaper. This link in German contains more information.]

The discontent of circular designs

You have two numbers +84% and -25%.

The textbook method to visualize this pair is to plot two bars. One bar in the positive direction, the other in the negative direction. The chart is clear (more on the analysis later).

But some find this graphic ugly. They don’t like straight lines, right angles and such. They prefer circles, and bends. Like PBS, who put out the following graphic that was forwarded to me by Fletcher D. on twitter:

Bending the columns is not as simple as it seems. Notice that the designer adds red arrows pointing up and down. Because the circle rounds onto itself, the sense of direction is lost. Now, readers must pick up the magnitude and the direction separately. It doesn’t help that zero is placed at the bottom of the circle.

Can we treat direction like we would on a bar chart? Make counter-clockwise the negative direction. This is what it looks like:

But it’s confusing. I made the PBS design worse because now, the value of each position on the circle depends on knowing whether the arrow points up or down. So, we couldn’t remove those red arrows.

The limitations of the “racetrack” design reveal themselves in similar data that are just a shade different. Here are a couple of scenarios to ponder:

1. You have growth exceeding 100%. This is a hard problem.
2. You have three or more rates to compare. Making one circle for each rate quickly becomes cluttered. You may make a course with multiple racetracks. But anyone who runs track can tell you the outside lanes are not the same distance as the inside. I wrote about this issue in a long-ago post (see here).

***

For a Trifecta Checkup (link), I'd also have concerns about the analytics. There are so many differences between the states that have required masks and states that haven't - the implied causality is far from proven by this simple comparison. For example, it would be interesting to see the variability around these averages - by state or even by county.

Bubble charts, ratios and proportionality

A recent article in the Wall Street Journal about a challenger to the dominant weedkiller, Roundup, contains a nice selection of graphics. (Dicamba is the up-and-comer.)

The change in usage of three brands of weedkillers is rendered as a small-multiples of choropleth maps. This graphic displays geographical and time changes simultaneously.

The staircase chart shows weeds have become resistant to Roundup over time. This is considered a weakness in the Roundup business.

***

In this post, my focus is on the chart at the bottom, which shows complaints about Dicamba by state in 2019. This is a bubble chart, with the bubbles sorted along the horizontal axis by the acreage of farmland by state.

Below left is a more standard version of such a chart, in which the bubbles are allowed to overlap. (I only included the bubbles that were labeled in the original chart).

The WSJ’s twist is to use the vertical spacing to avoid overlapping bubbles. The vertical axis serves a design perogative and does not encode data.  

I’m going to stick with the more traditional overlapping bubbles here – I’m getting to a different matter.

***

The question being addressed by this chart is: which states have the most serious Dicamba problem, as revealed by the frequency of complaints? The designer recognizes that the amount of farmland matters. One should expect the more acres, the more complaints.

Let's consider computing directly the number of complaints per million acres.

The resulting chart (shown below right) – while retaining the design – gives a wholly different feeling. Arkansas now owns the largest bubble even though it has the least acreage among the included states. The huge Illinois bubble is still large but is no longer a loner.

Now return to the original design for a moment (the chart on the left). In theory, this should work in the following manner: if complaints grow purely as a function of acreage, then the bubbles should grow proportionally from left to right. The trouble is that proportional areas are not as easily detected as proportional lengths.

The pair of charts below depict made-up data in which all states have 30 complaints for each million acres of farmland. It’s not intuitive that the bubbles on the left chart are growing proportionally.

Now if you look at the right chart, which shows the relative metric of complaints per million acres, it’s impossible not to notice that all bubbles are the same size.

All these charts lament the high prices charged by U.S. hospitals

A former student asked me about this chart from the New York Times that highlights much higher prices of hospital procedures in the U.S. relative to a comparison group of seven countries.

The dot plot is clearly thought through. It is not a default chart that pops out of software.

Based on its design, we surmise that the designer has the following intentions:

1. The names of the medical procedures are printed to be read, thus the long text is placed horizontally.
   
   
2. The actual price is not as important as the relative price, expressed as an index with the U.S. price at 100%. These reference values are printed in glaring red, unignorable.
   
   
3. Notwithstanding the above point, the actual price is still of secondary importance, and the values are provided as a supplement to the row labels. Getting to the actual prices in the comparison countries requires further effort, and a calculator.
   
   
4. The primary comparison is between the U.S. and the rest of the world (or the group of seven countries included). It is less important to distinguish specific countries in the comparison group, and thus the non-U.S. dots are given pastels that take some effort to differentiate.
   
   
5. Probably due to reader feedback, the font size is subject to a minimum so that some labels are split into two lines to prevent the text from dominating the plotting region.

***

In the Trifecta Checkup view of the world, there is no single best design. The best design depends on the intended message and what’s in the available data.

To illustate this, I will present a few variants of the above design, and discuss how these alternative designs reflect the designer's intentions.

Note that in all my charts, I expressed the relative price in terms of discounts, which is the mirror image of premiums. Instead of saying Country A's price is 80% of the U.S. price, I prefer to say Country A's price is a 20% saving (or discount) off the U.S. price.

First up is the following chart that emphasizes countries instead of hospital procedures:

This chart encourages readers to draw conclusions such as "Hospital prices are 60-80 percent cheaper in Holland relative to the U.S." But it is more taxing to compare the cost of a specific procedure across countries.

The indexing strategy already creates a barrier to understanding relative costs of a specific procedure. For example, the value for angioplasty in Australia is about 55% and in Switzerland, about 75%. The difference 75%-55% is meaningless because both numbers are relative savings from the U.S. baseline. Comparing Australia and Switzerland requires a ratio (0.75/0.55 = 1.36): Australia's prices are 36% above Swiss prices, or alternatively, Swiss prices are a 64% 26% discount off Australia's prices.

The following design takes it even further, excluding details of individual procedures:

For some readers, less is more. It’s even easier to get a rough estimate of how much cheaper prices are in the comparison countries, for now, except for two “outliers”, the chart does not display individual values.

The widths of these bars reveal that in some countries, the amount of savings depends on the specific procedures.

The bar design releases the designer from a horizontal orientation. The country labels are shorter and can be placed at the bottom in a vertical design:

It's not that one design is obviously superior to the others. Each version does some things better. A good designer recognizes the strengths and weaknesses of each design, and selects one to fulfil his/her intentions.

 

P.S. [1/3/20] Corrected a computation, explained in Ken's comment.

Revisiting global car sales

We looked at the following chart in the previous blog. The data concern the growth rates of car sales in different regions of the world over time.

Here is a different visualization of the same data.

Well, it's not quite the same data. I divided the global average growth rate by four to yield an approximation of the true global average. (The reason for this is explained in the other day's post.)

The chart emphasizes how each region was helping or hurting the global growth. It also features the trend in growth within each region.

 

This Excel chart looks standard but gets everything wrong

The following CNBC chart (link) shows the trend of global car sales by region (or so we think).

This type of chart is quite common in finance/business circles, and has the fingerprint of Excel. After examining it, I nominate it for the Hall of Shame.

***

The chart has three major components vying for our attention: (1) the stacked columns, (2) the yellow line, and (3) the big red dashed arrow.

The easiest to interpret is the yellow line, which is labeled "Total" in the legend. It displays the annual growth rate of car sales around the globe. The data consist of annual percentage changes in car sales, so the slope of the yellow line represents a change of change, which is not particularly useful.

The big red arrow is making the point that the projected decline in global car sales in 2019 will return the world to the slowdown of 2008-9 after almost a decade of growth.

The stacked columns appear to provide a breakdown of the global growth rate by region. Looked at carefully, you'll soon learn that the visual form has hopelessly mangled the data.

What is the growth rate for Chinese car sales in 2006? Is it 2.5%, the top edge of China's part of the column? Between 1.5% and 2.5%, the extant of China's section? The answer is neither. Because of the stacking, China's growth rate is actually the height of the relevant section, that is to say, 1 percent. So the labels on the vertical axis are not directly useful to learning regional growth rates for most sections of the chart.

Can we read the vertical axis as global growth rate? That's not proper either. The different markets are not equal in size so growth rates cannot be aggregated by simple summing - they must be weighted by relative size.

The negative growth rates present another problem. Even if we agree to sum growth rates ignoring relative market sizes, we still can't get directly to the global growth rate. We would have to take the total of the positive rates and subtract the total of the negative rates.  

***

At this point, you may begin to question everything you thought you knew about this chart. Remember the yellow line, which we thought measures the global growth rate. Take a look at the 2006 column again.

The global growth rate is depicted as 2 percent. And yet every region experienced growth rates below 2 percent! No matter how you aggregate the regions, it's not possible for the world average to be larger than the value of each region.

For 2006, the regional growth rates are: China, 1%; Rest of the World, 1%; Western Europe, 0.1%; United States, -0.25%. A simple sum of those four rates yields 2%, which is shown on the yellow line.

But this number must be divided by four. If we give the four regions equal weight, each is worth a quarter of the total. So the overall average is the sum of each growth rate weighted by 1/4, which is 0.5%. [In reality, the weights of each region should be scaled to reflect its market size.]

***

tldr; The stacked column chart with a line overlay not only fails to communicate the contents of the car sales data but it also leads to misinterpretation.

I discussed several serious problems of this chart form: 

• stacking the columns make it hard to learn the regional data
  
  
• the trend by region takes a super effort to decipher
  
  
• column stacking promotes reading meaning into the height of the column but the total height is meaningless (because of the negative section) while the net height (positive minus negative) also misleads due to presumptive equal weighting
  
  
• the yellow line shows the sum of the regional data, which is four times the global growth rate that it purports to represent

 

***

PS. [12/4/2019: New post up with a different visualization.]

How to read this cost-benefit chart, and why it is so confusing

Long-time reader Antonio R. found today's chart hard to follow, and he isn't alone. It took two of us multiple emails and some Web searching before we think we "got it".

 

Antonio first encountered the chart in a book review (link) of Hal Harvey et. al, Designing Climate Solutions. It addresses the general topic of costs and benefits of various programs to abate CO2 emissions. The reviewer praised the "wealth of graphics [in the book] which present complex information in visually effective formats." He presented the above chart as evidence, and described its function as:

policy-makers can focus on the areas which make the most difference in emissions, while also being mindful of the cost issues that can be so important in getting political buy-in.

(This description is much more informative than the original chart title, which states "The policy cost curve shows the cost-effectiveness and emission reduction potential of different policies.")

Spend a little time with the chart now before you read the discussion below.

Warning: this is a long read but well worth it.

 

***

 

If your experience is anything like ours, scraps of information flew at you from different parts of the chart, and you had a hard time piecing together a story.

What are the reasons why this data graphic is so confusing?

Everyone recognizes that this is a column chart. For a column chart, we interpret the heights of the columns so we look first at the vertical axis. The axis title informs us that the height represents "cost effectiveness" measured in dollars per million metric tons of CO2. In a cost-benefit sense, that appears to mean the cost to society of obtaining the benefit of reducing CO2 by a given amount.

That's how far I went before hitting the first roadblock.

For environmental policies, opponents frequently object to the high price of implementation. For example, we can't have higher fuel efficiency in cars because it would raise the price of gasoline too much. Asking about cost-effectiveness makes sense: a cost-benefit trade-off analysis encapsulates the something-for-something principle. What doesn't follow is that the vertical scale sinks far into the negative. The chart depicts the majority of the emissions abatement programs as having negative cost effectiveness.

What does it mean to be negatively cost-effective? Does it mean society saves money (makes a profit) while also reducing CO2 emissions? Wouldn't those policies - more than half of the programs shown - be slam dunks? Who can object to programs that improve the environment at no cost?

I tabled that thought, and proceeded to the horizontal axis.

I noticed that this isn't a standard column chart, in which the width of the columns is fixed and uneventful. Here, the widths of the columns are varying.

***

In the meantime, my eyes are distracted by the constellation of text labels. The viewing area of this column chart is occupied - at least 50% - by text. These labels tell me that each column represents a program to reduce CO2 emissions.

The dominance of text labels is a feature of this design. For a conventional column chart, the labels are situated below each column. Since the width does not usually carry any data, we tend to keep the columns narrow - Tufte, ever the minimalist, has even advocated reducing columns to vertical lines. That leaves insufficient room for long labels. Have you noticed that government programs hold long titles? It's tough to capture even the outline of a program with fewer than three big words, e.g. "Renewable Portfolio Standard" (what?).

The design solution here is to let the column labels run horizontally. So the graphical element for each program is a vertical column coupled with a horizontal label that invades the territories of the next few programs. Like this:

The horror of this design constraint is fully realized in the following chart, a similar design produced for the state of Oregon (lifted from the Plan Washington webpage listed as a resource below):

In a re-design, horizontal labeling should be a priority.

 

***

Realizing that I've been distracted by the text labels, back to the horizontal axis I went.

This is where I encountered the next roadblock.

The axis title says "Average Annual Emissions Abatement" measured in millions metric tons. The unit matches the second part of the vertical scale, which is comforting. But how does one reconcile the widths of columns with a continuous scale? I was expecting each program to have a projected annual abatement benefit, and those would fall as dots on a line, like this:

Instead, we have line segments sitting on a line, like this:

Think of these bars as the bottom edges of the columns. These line segments can be better compared to each other if structured as a bar chart:

Instead, the design arranges these lines end-to-end.

To unravel this mystery, we go back to the objective of the chart, as announced by the book reviewer. Here it is again:

policy-makers can focus on the areas which make the most difference in emissions, while also being mindful of the cost issues that can be so important in getting political buy-in.

The primary goal of the chart is a decision-making tool for policy-makers who are evaluating programs. Each program has a cost and also a benefit. The cost is shown on the vertical axis and the benefit is shown on the horizontal. The decision-maker will select some subset of these programs based on the cost-benefit analysis. That subset of programs will have a projected total expected benefit (CO2 abatement) and a projected total cost.

By stacking the line segments end to end on top of the horizontal axis, the chart designer elevates the task of computing the total benefits of a subset of programs, relative to the task of learning the benefits of any individual program. Thus, the horizontal axis is better labeled "Cumulative annual emissions abatement".

 

Look at that axis again. Imagine you are required to learn the specific benefit of program titled "Fuel Economy Standards: Cars & SUVs".  

This is impossible to do without pulling out a ruler and a calculator. What the axis labels do tell us is that if all the programs to the left of Fuel Economy Standards: Cars & SUVs were adopted, the cumulative benefits would be 285 million metric tons of CO2 per year. And if Fuel Economy Standards: Cars & SUVs were also implemented, the cumulative benefits would rise to 375 million metric tons.

***

At long last, we have arrived at a reasonable interpretation of the cost-benefit chart.

Policy-makers are considering throwing their support behind specific programs aimed at abating CO2 emissions. Different organizations have come up with different ways to achieve this goal. This goal may even have specific benchmarks; the government may have committed to an international agreement, for example, to reduce emissions by some set amount by 2030. Each candidate abatement program is evaluated on both cost and benefit dimensions. Benefit is given by the amount of CO2 abated. Cost is measured as a "marginal cost," the amount of dollars required to achieve each million metric ton of abatement.

This "marginal abatement cost curve" aids the decision-making. It lines up the programs from the most cost-effective to the least cost-effective. The decision-maker is presumed to prefer a more cost-effective program than a less cost-effective program. The chart answers the following question: for any given subset of programs (so long as we select them left to right contiguously), we can read off the cumulative amount of CO2 abated.

***

There are still more limitations of the chart design.

• We can't directly read off the cumulative cost of the selected subset of programs because the vertical axis is not cumulative. The cumulative cost turns out to be the total area of all the columns that correspond to the selected programs. (Area is height x width, which is cost per benefit multiplied by benefit, which leaves us with the cost.) Unfortunately, it takes rulers and calculators to compute this total area.
  
  
• We have presumed that policy-makers will make the Go-No-go decision based on cost effectiveness alone. This point of view has already been contradicted. Remember the mystery around negatively cost-effective programs - their existence shows that some programs are stalled even when they reduce emissions in addition to making money!
  
  
• Since many, if not most, programs have negative cost-effectiveness (by the way they measured it), I'd flip the metric over and call it profitability (or return on investment). Doing so removes another barrier to our understanding. With the current cost-effectiveness metric, policy-makers are selecting the "negative" programs before the "positive" programs. It makes more sense to select the "positive" programs before the "negative" ones!

***

In a Trifecta Checkup (guide), I rate this chart Type V. The chart has a great purpose, and the design reveals a keen sense of the decision-making process. It's not a data dump for sure. In addition, an impressive amount of data gathering and analysis - and synthesis - went into preparing the two data series required to construct the chart. (Sure, for something so subjective and speculative, the analysis methodology will inevitably be challenged by wonks.) Those two data series are reasonable measures for the stated purpose of the chart.

The chart form, though, has various shortcomings, as shown here.  

***

In our email exchange, Antonio and I found the Plan Washington website useful. This is where we learned that this chart is called the marginal abatement cost curve.

Also, the consulting firm McKinsey is responsible for popularizing this chart form. They have published this long report that explains even more of the analysis behind constructing this chart, for those who want further details.

Who is a millennial? An example of handling uncertainty

I found this fascinating chart from CNBC, which attempts to nail down the definition of a millennial.

It turns out everyone defines "millennials" differently. They found 23 different definitions. Some media outlets apply different definitions in different items.

I appreciate this effort a lot. The design is thoughtful. In making this chart, the designer added the following guides:

• The text draws attention to the definition with the shortest range of birth years, and the one with the largest range.
• The dashed gray gridlines help with reading the endpoints of each bar.
• The yellow band illustrates the so-called average range. It appears that this average range is formed by taking the average of the beginning years and the average of the ending years. This indicates a desire to allow comparisons between each definition and the average range.
• The bars are ordered by the ending birth year (right edge).

The underlying issue is how to display uncertainty. The interest here is not just to feature the "average" definition of a millennial but to show the range of definitions.

***

In making my chart, I apply a different way to find the "average" range. Given any year, say 1990, what is the chance that it is included in any of the definitions? In other words, what proportion of the definitions include that year? In the following chart, the darker the color, the more likely that year is included by the "average" opinion.

I ordered the bars from shortest to the longest so there is no need to annotate them. Based on this analysis, 90 percent (or higher) of the sources list 19651985 to 1993 as part of the range while 70 percent (or higher) list 19611981 to 1996 as part of the range.