One Way to Mislead With Charts

Today I'd like to take a break from my recent topics of discussion and look at an example of why people shold be skeptical of the messaging by global warming advocates. This post isn't about science. I'm not going to argue about any facts or theories. I'm not going to question or put forth facts or evidence.

None of those things matter today. Regardless of what one believes about global warming, everyone should be able to agree on a basic principle: Results should be presented in an accurate manner that does not create a misleading impression of what the results show. And based upon that principle, everyone should be able to agree this display is rubbish:

I'm not questioning the data used to make this display. The data doesn't matter today. What matters today is the data is being displayed in a misleading manner.

That chart shows global temperatures over the last 150 or so years in a spiral pattern. Because there are a couple things to unpack in it, it might help to examine a different graphic first. Consider the one found in this link (sorry, I couldn't figure out how to embed it):

That shows the evolution of ice in the Arctic and Antarctic areas. The two smaller spirals in it show each region's individual ice levels while the larger spiral shows the total ice level. Toward the end of the spiral, there's a dramatic decline. Or at least, it appears dramatic. Look at these two circles:

How much of a difference do you think there is in size between these two circles? It's double. More specifically, the circle on the left is 1.96 times the size of the circle on the right. That's because these represent pizzas.

If you have to decide between a 10 inch and 14 inch pizza, you might think four inches isn't that much. You'd be wrong. A 14 inch pizza is twice as large as a 10 inch pizza. That's because the area of a circle increases by the square of the radius. A circle with a radius of five inches (10 inch diameter) has an area of 3.14 * 5^2, or 78.5 square inches. A circle with a seven inch radius (14 inch diameter) is 3.14 * 7^2, or 154 square inches.

With this in mind, look at the two spiral charts above. Each month of the year has its data plotted at a certain angle. If the data were the same in each month of a year, you'd have a perfect circle. As the data values change, the radius for that particular point changes as the radius is determined by the data.

Now remember what we saw with the pizzas. Going from a seven inch radius to a five inch radius is a change of ~30%. That's not what the eye sees though. When we see a circle or spiral, our eyes don't focus on the radius of that circle. Our eyes focus on the area. As we saw with the pizzas, a change from a seven inch radius to a five inch radius reduces the area by ~50%.

Plotting any series with a trend in a spiral pattern like this will exaggerate the apparent rate of change. Consider this particularly bad example:

While some credit is deserved for the graph having radius markers on it, the reality is people looking at that animation will not focus on the radius markers. What they will see is the enormous change in area. They will see temperatures rising to 100 times the original levels when in reality the (projected future) temperatures only increases a twenieth of that. That means the visual impact is exaggerated by ~2000%.

That's not the only problem though. Remember how I said I wanted to look at the ice level charts first because they were simpler? Those animations were simpler because the amount of ice cannot drop below 0. Temperatures can. You can see that in the temperature spiral as some past temperatures are inside the 0C line.

Here's the thing, circles can't have a radius smaller than one. To create these temperature spirals with values below zero, the center of the circle cannot represent zero. It has to represent some other number. What number though? In these charts the number is -1, but why? Why use -1 instead of -2 or -0.5? There is no inherent reason one number is a better centerpoint for the circle than another.

This is important because it affects the area of the circle. Starting at -1 means 0C is 1, 1C is 2, 2C is 3 and so forth. This means the area goes 3.14, 12.56, 28.27, 50.27, 78.54, 113.10. If we set the centerpoint of the circle at -0.5 instead of -1, those values would change to 0.79, 7.07, 19.63, 38.48, 63.62, 95.03. Take note of how 113.10/3.14 is 36.02 while 95.03/0.79 is is 120.29. That shows how much the visual impact is exaggerated by plotting this data in a spiral is affected by the arbitrary choice of what temperature value to set the center of the circle at.

The numbers I've discussed come from basic geometry. Climate scientists understand these numbers every bit as well as I do. They also understand that humans typically focus on the area of a circle rather than the radius. Why then are charts like these heavily promoted and praised as wonderful tools informing people about global warming?

I don't care to guess what the answer to that question is. It doesn't matter. I don't think climate scientists are looking at these plots and cackling as they think about how good they are at deceiving people. I wouldn't care if they were. What I care about is rubbish charts like these are accepted by the scientific community that is claiming to inform the public about global warming.

Guess what guys? If you post misleading charts, people should distrust you. If you want to use this approach to displaying your data, that's fine. It's okay to plot data in a way people won't instinctively grasp. You just need to clearly explain to them how they should interpret your chart. Or you could do this:

The left chart shows how you'd plot data proportional to the radius. The right chart shows how you'd plot data proportional to the area. It would be easy for the people who have made these spiral charts to show their spirals on both of these. They could explain to people why the resulting animations look so different. That would let their audience see the spiral animations and actually understand what they meant.

I understand many people won't care about this. Doing as I suggest wouldn't be "sexy." It wouldn't create a compelling case. What would it would do is be fair and accurate. The spiral charts being created by climate scientists and promoted by global warming advocates are neither.


  1. Context probably is the most important information conveyed by charts and graphs. All of the unstated premises, limitations, boundaries, and caveats are necessary to interpret the nuggets of data. When context is left out or hidden, it leaves the recipient to fill in the blanks with guesswork, which certainly will not be as helpful as a careful explanation of how to understand the data. You make an excellent point that imposing a linear scale on the areal characteristics of concentric circles is shoddy work at best.

  2. Thanks Gary. I do think these sorts of animations could be useful if a person was properly prepared on how to interpret them. This is a way of showing a lot of information in a fairly concise manner. I like the approach for that reason. I also think a rescaled version of the charts would have its own problems as it would create a misleading impression when people compared individual points on the spiral. That means neither approach is "right." Both have their problems.

    That's okay. Visualizations are often imperfect. That's why we explain them. That's why we provide details and information, or context as you put it, with them - so people can properly interpret what they are seeing. Nobody is doing that. That's why (in my opinion, at least) they shouldn't be trusted. It's not that i think most climate scientists and global warming advocates have nefarious desires. I'm sure there are any number of reasons for this sort of behavior other than, "They're evil!"

    I don't care what those explanations are though. To me, you shouldn't be trusted so long as you present misleading information. I don't care why you're doing it.

  3. One detail, lots of places have medium and large at 12 and 14, though the smaller indy shops frequently do 16.

    I think they do something similar with IPCC charts, like having proxies disappear under a red temperature curve.

  4. Hi Brandon,

    Thank you for featuring my spiral in this well written blog post.

    Here's some code to embed my spiral: <iframe class="imgur-embed" width="100%" height="1000" frameborder="0" src=""></iframe> (if that html gets filtered out of this comment then you can simply use this

    I agree that it is difficult to interpret changes in area of a circle and your example with the pizzas is spot on. Tufte in his superb book The Visual Display of Quantitative Information goes into great detail about the evils of using area to represent a linear value.

    A steadily increasing value plotted on a spiral is certainly dramatic and the exponential increase in "ink" (to steal terminology from Tufte) has a massive psychological effect on the data represented. Unfortunately you completely missed that the sea ice area is reducing in value which means the amount of ink is exponentially decreasing. Thus the psychological impact should also exponentially reduce in impact. This is clearly not so as the resulting spiral still shows a dramatic reduction.

    Me plotting this data on a spiral, in theory, should be a stupid idea if I really wanted to mislead my audience.

    Looking forward to Friday, it's pizza night at our place!

    Kev Pluck.

  5. Kev Pluck, thanks for the comment. I'm not sure if WordPress will let me embed iframes in my posts, but I'll keep the option in mind.

    I'm glad to hear we agree on much, but I'm afraid I don't understand why there is any disagreement. You say:

    Unfortunately you completely missed that the sea ice area is reducing in value which means the amount of ink is exponentially decreasing. Thus the psychological impact should also exponentially reduce in impact.

    But I don't see any reason that should be true. The psychological impact of a display is not based upon its absolute size but its relative size. It is how the various components of the display relate to one another which determines the psychological impact. The greater the difference in size between various components, the greater the impact.

    The problem here is not how big or small any part of the spiral is. The problem is how big or small each part of the spiral is in relation to each other part (and the rest of the display). This is true regardless of which direction the difference goes in.

    Exaggerating the rate of change is wrong. It doesn't stop being wrong because the rate of change is negative. The rate being negative just means you're going to create the impression things are declining faster than they actually are - increasing the apparent cause for concern.

    If you convince people something is vanishing faster than it actually is, the psychological impact will be greater than if you gave an accurate portrayal of things.

  6. The spiral displays an area. Sea ice area is an area. Where is the exponential decay coming from?

  7. MikeN, that a measurement measures area does not mean its numerical values should be represented in a graphic by area. If it did, practically every chart showing sea ice area would be wrong.

    The misleading nature of the sea ice spiral discussed in this post is how it makes a decline from say ten million square meters to nine million seem much smaller than the decrease of two million to one million. Using a spiral causes the latter drop appear four times faster than the former even though both are exactly the same.

    For a simple thought experiment, imagine you have a yard you want to mow. Create a spiral chart showing how much of it you've mowed over time. Even if you maintain a constant rate of mowing, your chart will make it look like you sped up as you went.

  8. (I don't appear to be getting notifications for replies so only spotting updates when I go ego googling).

    My spiral definitely has the distortions you describe, in fact I'm working on a new visualisation that show Arctic (and eventually Antarctic) sea ice area in a more direct manner and hopefully in a more compelling way than the spiral.

    The amount of "ink", i.e. the number of pixels, inscribed in an arc as the radius increases grows exponentially dramatically effecting the visual impact (a term I perhaps should have used instead of "psychological impact"). Conversely as the radius decreases the amount of ink shrinks exponentially with an equivalent effect on the visual impact.

    Bear in mind that there is no distortion of the linear distance of a point from the center. The only distortion is the length of the arc inscribed over the same angle at different radii.

    So my point is that as the radius reduces it does not "exaggerate" the rate of change but actually lessens the rate of change which just shows that the actual rate of reduction in ice area is so great that using a visualisation that lessons the rate of change still has a major impact.

    Here is the same data plotted in unscaled cartesian coordinates showing the true rate of change:

    Absolutely frightening any way you look at it.

  9. (I can't ego Google. I'd be too afraid of what I find. No clue why you wouldn't get update notifications. I'm not sure if this site uses them or not.)

    It sounds like we mostly agree, From what I can tell, we agree up to the point reducing the amount of "ink" means you reduce the apparent rate of change. I agree with that. My problem is not with the apparent rate of change. My problem is with the mismatch between the apparent rate of change and the actual rate of change. By making it appear the rate of change is smaller than it is, you make the actual rate of change seem faster than it is.

    As a test, make a spiral that goes all the way to zero. Trace the spiral with your finger (or just eyes) and try to estimate what the numerical is as you go. Most people will give numbers in the form of a negative exponential. They will give a faster rate of change for the outer portions of the spiral. As you say, they will think the rate of change is slowing down. At the same time, they will think they are "closer" to zero than they actually are. If 25% of the sea ice area is left, they'll think only ~5% is left. That can easily make things seem more alarming than they actually are.

    What's happening here is the exaggeration of the rate of change in the outer arcs makes it appear there is less room for change in the inner arcs. This makes it seems you are closer to zero than you actually are. That is how the actual rate of change gets exaggerated even though the apparent rate of change is reduced. If you use a coordinate system scaled to account for the bias introduced by plotting a linear value in an areal manner, the apparent rate of change in the inner segments will increase but it won't seem you're as "close" to zero.

    Here is the same data plotted in unscaled cartesian coordinates showing the true rate of change:

    Absolutely frightening any way you look at it.

    For what it's worth, I don't find that particular chart frightening at all. The colors don't do it for me. Intuitively, my mind doesn't see a sense of progression over the years. I couldn't tell you which year was more recent than the next without the legend (or my personal knowledge of the subject). 2016 has the most notable line because it is an outlier and is so thick, but after that line, my eyes jump to 2012's because its thicker than the lines for 2013-2015.

    That doesn't frighten me. It just confuses me. For me, the inconsistent thickness of the lines completely ruins any sense of temporal progression that might otherwise be indicated by the change in colors from year to year. Without a legend, I would never be able to tell you which year was which (save for the last two).

  10. It does sound like we mostly agree. I'm afraid that I can't agree with your position of the rate of change though.

    As for Wipneus's chart I agree that it is a mess of colours and line thicknesses which is why I chose to reproduced it.

    Calling 2016/17 an outlier is dismissing the significance of the truly monumental shift in sea ice area. This is an outlier: mere noise that can be easily filtered out of data. But when literally hundreds of samples which have been consistently over 5 standard deviations outside the normal then it is not a mere outlier, it is real and cannot be ignored.

  11. Kev Pluck, I didn't intend to minimize the change by calling it an outlier. I was just recognizing that it is an outlier, and as such, it stands out in the chart. Sometimes outliers are anomalies caused by something which isn't relevant to one's concerns. Other times, they are essential pieces of data that indicate something important. Simply calling something an outlier doesn't indicate whether it is important or not, though I can understand why someone might misunderstand what is meant.

    All I meant by calling that data an outlier is that it stands out from the rest. Most of the other lines in the chart blur together so much a casual viewer won't glean any real information from them. 2016 is the only (full) year whose data stands out in the chart. That's all I was saying.

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