How a decorated graph can mislead an honest reader

0:00Here is our column graph of the seven summits. Correct, plain, and slightly dull. So people decorate them. Colour, shapes, pictures of the thing being measured. The book has a word for the result. An infographic — short for information graphics. And the reason for making one is perfectly good. People look at an infographic. They scroll past a bar chart. A graph that nobody reads has answered nothing, which is exactly where we left off last time.

0:34So this is not a warning against decorating your data. It is a warning about what decoration can do while you are not watching. Watch the first redraw, one change at a time. Step one. Each rectangle becomes a triangle — a little mountain shape, pointed at the top. Step two. Colour, a different one for each summit. Step three. Labels on every peak. The mountain, its height, its continent.

1:06And it looks good. Genuinely better than what we started with. Every apex is at exactly the right height. Nobody has faked a number. The scale is still there along the side. And something real was gained, which is worth saying before we take it apart. The shapes now look like the thing they stand for. A mountain is drawn as a mountain. You no longer have to remember what the picture is about. It tells you.

1:38The labels mean you do not have to trace across to a scale to get a height — it is printed on the peak. So this picture is easier to read, easier to remember, and more likely to be looked at in the first place. All three of those matter. Keep them in mind, because we are about to find something wrong. Now measure the bottom of each triangle. Not the height — the width.

2:06And they are not the same. The taller triangles are wider. The short ones are narrow. Nobody decided that. It happened while the shapes were being drawn to look nice. But look at what it does. Height carries a meaning here — it is the mountain's height, declared by the scale. Width now varies too. And nothing anywhere says what width means. So the picture has acquired a second signal, and no key for it.

2:37Which is exactly why a bar graph makes all its bars the same width. That rule has looked like fussiness for two videos now. Here is what it was for. In a bar graph, width is held constant so that it carries nothing. It cannot say anything, because it never changes. So length is the only thing varying, and length is the only thing meaning anything. One varying quantity, one meaning.

3:06The moment a second property starts changing alongside it, your reader has two signals and one key. And they will read both. Nobody can help reading both. So put the claim into words, because that is when it becomes testable. The picture is saying: taller mountains are broader mountains. Is that true? Well — is it? Is Everest wider at the base than Kilimanjaro? I do not know. You do not know. And more to the point, the table does not know, because the table has one column of heights and nothing about width at all.

3:47So the picture is asserting something that its own data cannot support. It might be true. It might be nonsense. And notice nobody lied. The book's own word for how this happens is accidentally. Somebody made the drawing prettier and it picked up a claim on the way. There is something the book does not compute here, and it is worth doing, because it shows how big this effect really is.

4:17If the base grows in proportion to the height, then the whole triangle scales both ways at once. And when a shape gets both taller and wider by the same factor, its area grows by that factor squared. So take Everest against Kosciuszko. Everest is eight thousand eight hundred and forty-eight; Kosciuszko is two thousand two hundred and twenty-eight. Everest is about four times as tall. But four squared is sixteen. Everest's triangle covers nearly sixteen times the ink.

4:55Sixteen times the area, for a mountain four times the height. And your eye reads area. It does not carefully attend to apex height and discard everything else — it takes in how much of the page a shape occupies. So the exaggeration is not a small side effect of a design choice. It is roughly four times over. Now the second redraw, and it goes further. The seven separate triangles dissolve into one continuous painted mountain range, with the seven summits picked out along it.

5:34Each is still labelled. Name, continent, height. It is a genuinely beautiful picture. But look at what has happened around the seven peaks. There is now a whole landscape there — ridges, valleys, slopes joining one summit to the next. None of that is in the data. It has been invented, because a range has to look like a range. And these seven mountains are on seven different continents. There is no range. They are thousands of kilometres apart.

6:08The picture has put them next to each other, which is a claim about the world that is simply not true. Now here is the test the book actually sets, and it is a good one because it is answerable. Look at Everest on that range, and look at Elbrus. What is your impression? Everest looks about twice as tall. Trust that for a moment. It is a reasonable reading of the picture — that is genuinely what the drawing shows.

6:39And then the book asks you one arithmetic question, and only one. What is five thousand six hundred and forty-two, doubled? Do it. Do not carry on until you have. Five thousand six hundred and forty-two, doubled, is eleven thousand two hundred and eighty-four. And Everest is eight thousand eight hundred and forty-eight. Which is nowhere near eleven thousand. It is two thousand four hundred and thirty-six metres short. So Everest is not twice Elbrus. Everest is about one and a half times Elbrus — one point five seven, if you divide it out.

7:22The picture said two. The numbers say one and a half. That is an overstatement of about twenty-eight per cent. And notice what settled it. Not judgement, not taste, not an argument about whether the picture is nice. One multiplication. That is the defence, and it is available to anybody. Read a number off the picture, then go and check it against the table. One more thing, and it is small, and that is precisely why it is worth showing.

7:57Kilimanjaro's height. On the table it is five thousand eight hundred and ninety-five metres. On the triangle picture, the same — five eight nine five. On the painted range, it is printed as five thousand eight hundred and ninety-two. Three metres. A tiny slip, and it changes nothing about any conclusion. But think about where it turned up. Not in the table, not in the plain graph — in the most decorated version, the one that took the most work to make.

8:32That is the pattern. The more a picture is handled, redrawn and relabelled, the more chances a number has to drift. And a three-metre error is completely invisible on a drawing. You could never see it. The only thing that catches it is checking the picture against the table. So the chapter closes with a warning, and the important thing about it is that it points in two directions. The first is obvious. Do not be misled by somebody else's picture. When a data picture makes an impression on you, test the impression before you repeat it.

9:13The second is the one people skip. Do not mislead others with your own. Nobody in this example was dishonest. The person drawing those triangles was trying to make something clear and attractive. The extra claim arrived uninvited. Which means you cannot rely on your own good intentions. You have to check your own picture, the same way you would check somebody else's. So, a short list for any decorated graph. Is the scale still there and still even? Is anything other than length varying — width, area, shape? Do the labels match the table they came from?

9:57And the one that catches most of it: pick two values, read the picture's claim about them, and do the arithmetic. A graph either preserves the quantities or it does not. That is not a matter of taste. It is decidable — and you decide it by computing.