What makes a data picture clear and worth looking at

0:00Here is a table, and everything in it is correct. The tallest mountain on each of the seven continents, with its height in metres. Everest, eight thousand eight hundred and forty-eight. Aconcagua, six thousand nine hundred and sixty-two. Denali, six one nine four. Kilimanjaro, five eight nine five. Elbrus, five six four two. Vinson Massif, four eight nine two. And Kosciuszko, two thousand two hundred and twenty-eight. Seven rows, seven right answers, nothing wrong with any of it.

0:38And as an instrument it is close to useless. This video is about why. The book asks two questions about that table, and you should try them before we go on. First: how much taller is Everest than the Australian summit? Your eye finds two rows, holds two four-digit numbers, and subtracts. Six thousand six hundred and twenty. Doable — but notice you did arithmetic, and hunted for the rows first.

1:09Second question, and this one is worse. Are Denali and Kilimanjaro very different in height? Six one nine four against five eight nine five. Two numbers of the same length on the page. Big difference, or small? You genuinely cannot tell by looking. You have to subtract, and it comes to two hundred and ninety-nine metres. Which is small — under five per cent of Denali. Nothing in the table told you that. It made you compute it.

1:43So draw it instead. The first decision is one we already know how to make. What is one unit of length worth? The largest value is eight thousand eight hundred and forty-eight, so mark the scale in steps of a thousand metres, up to nine thousand. And here the book gives the scale a second job, which is worth naming. The scale is how you make a picture fit the space it has been given.

2:12Same decision, two reasons. One arithmetic — the heights must be computable. One practical — the drawing must fit the page. Now turn each row into a box running across the page. Everest, a box just short of nine thousand. Aconcagua, a bit under seven. Denali, just over six. Kilimanjaro, Elbrus, Vinson, and then Kosciuszko — a short one, a little over two thousand. Nothing has been added. Same seven numbers, same scale, nothing rounded.

2:50All that changed is that a number is now a length. And look what you get for nothing. Rank the seven, tallest to shortest. On the table that was a sorting job. Here you read down the ends of the boxes. Go back to the first question. You still have to subtract to get six thousand six hundred and twenty — the picture does no arithmetic for you. But now go to the second question, and it has evaporated.

3:21Are Denali and Kilimanjaro very different? Look at them. Their ends are practically level. That two hundred and ninety-nine metres is under a third of one scale step. On the picture it is a sliver. The table forced you to compute an answer you can now simply see. That is what drawing bought. But there is something better here, and the book does not ask for it. Look at the seven boxes as a group, and a shape appears.

3:54Everest, alone at the top. Then five mountains bunched close together. Then Kosciuszko, alone at the bottom. Check it against the numbers. The middle five run from four thousand eight hundred and ninety-two to six thousand nine hundred and sixty-two — a spread of about two thousand across all five. Everest stands one thousand eight hundred and eighty-six metres above the top of that group. And Kosciuszko sits two thousand six hundred and sixty-four metres below the bottom of it.

4:31So the two biggest gaps in the list are the first and the last. The middle five are, by comparison, much of a muchness. That is a real fact about the seven summits, and it sat in the table the whole time. Nobody could see it there. One picture, and it is the first thing you notice. Now the same graph, turned a quarter turn, so the boxes stand up instead of lying down.

5:00Watch the picture rotate, because the point is that nothing about the data moves with it. Same seven values. Same steps of a thousand metres. The bars are the same lengths they were. Two small things differ. The names sit along the bottom instead of down the side, and the marked line runs a little further — to ten thousand rather than nine. No height has changed, no comparison has changed. If Denali and Kilimanjaro were level before, they are level now.

5:35The only thing that has been altered is which way your eye travels to read it. And that upright arrangement has a name. A column graph. Which is a good name, and worth a second, because it comes from buildings. A column in a building is the upright pillar that holds up a roof. Vertical, standing on the ground, carrying something above it. The bars in this graph do exactly that. They stand on the horizontal line and rise.

6:08So a column graph is not a new kind of graph. It is a bar graph whose bars happen to be vertical. Same rules, same scale, same everything. Just stood upright and given a name. Now the argument the chapter is really making. For this data the upright version is better — and for a reason that has nothing to do with taste. The thing being measured is height. And height, out in the world, is measured upward from the ground.

6:41A mountain goes up. So a bar that also goes up is running the same way as the thing it stands for. Which means the reader does not have to translate anything. A taller mountain is a taller bar, and taller means the same thing in both. Draw the same data lying flat and it still works — but the reader silently converts every time: longer means taller. Small cost, paid on every single bar, by every single reader.

7:15And then the other half of the rule, which follows immediately. Some quantities do not go up at all. They lie flat. The distance between two cities. The length of a river. None of those is measured upward from anything. They run across the ground. So draw them across. Horizontal bars, and again the picture runs the same way as the world. That is the whole rule, in one line. Measured upward, vertical bars. Measured along the ground, horizontal ones.

7:51It is not a law, and nobody marks you wrong. It removes one small piece of work from your reader. The book gives you two to try, and they are chosen so that the rule decides them. One. The height of the tallest person in each class in your school. Go and measure, then graph it. That is a height. Heights are measured upward. So — vertical bars, and you did not have to think about it.

8:20Two. The longest river on each continent. You will have to look those up. A river's length lies along the ground. It does not go up. So — horizontal bars. Two data sets, two directions, and the same one-line rule settled both of them. You might reasonably ask whether any of this is mathematics. It sounds like a lesson in drawing neatly. So here is the case for it, and I think it holds.

8:51A graph is not a decoration attached to a result. The graph is how the result gets from you into somebody else's head. Collect the data honestly, table it correctly, compute every height right — then draw something nobody can read, and you have answered nothing. The seven-summit table was completely accurate and almost useless. That is the whole demonstration. Accuracy and usefulness are two different properties, and you have to get both.

9:25And notice everything we did was rearrangement. No number changed, nothing added, nothing dressed up. We moved seven values around until they could be read. Which raises an obvious question for next time. What happens when somebody does start adding things — colours, pictures, decoration — to a graph?