Dot plots: seeing spread and clustering at a glance

0:00Onions, two towns, twelve months. Twenty four prices in a grid. The question is simple. Which town is dearer? Five people look at the same table and answer five different ways. One finds the highest price anywhere and it is in the south. One adds the columns: four hundred and fifty eight against four hundred and fifty, so the north. One counts the expensive months. One goes month by month and finds the north dearer in six, the south in five, and one month tied.

0:37And one measures the gap between each town's cheapest and dearest: thirty five against forty three. Five honest yardsticks. No agreement. But look at what is actually going wrong here, because it is not that anybody is being unfair. It is that nobody can see the data. Twenty four numbers in a grid is not a picture of anything. Your eye cannot hold twelve numbers at once, so you reach for a summary, and every summary throws almost all of it away.

1:13So before summarising anything, try looking at it. Draw a line and number it, in rupees. Now take the prices one at a time and drop each one where it belongs. Twenty five. Twenty four. Twenty six. Twenty eight. Not a bar. Not a column. One dot, one price. Keep going to the end of the year, and there is the north town, all twelve months, as a picture. That is a dot plot, and it is nothing more than the table poured onto a line.

1:47Every dot is one month. Nothing has been added and nothing has been calculated. Now do the south town, and something happens that did not happen before. Forty two, in July. And forty two again, in December. The second dot has nowhere to go, so it goes on top. The stack grows upwards. And here is the mistake almost everybody makes on their first dot plot. That stack of two is not eighty four rupees. Eighty four is not a price. Nobody paid eighty four.

2:25The height of a stack is a count. It says two months, at forty two rupees each. Sideways is the value. Upwards is how many. Two directions, two completely different jobs. One more thing before reading anything off it. Look at where the line starts. It starts at ten. Not at zero. And there is a reason: nothing costs less than ten or more than sixty, so a stretch from zero to ten would be empty line.

2:58Cutting it out is a choice, and it is a defensible one. Eighty six percent of what is drawn now carries data, against about seventy percent if it began at zero. But it is not free, and this is the part people skip. On the shorter line, the same five rupee difference is drawn one point two times wider. Nothing is being hidden. But every gap now looks bigger than it would have, and you should know that before you say one town is far dearer than the other.

3:34Now, what did that cost us? Because something was definitely lost. Look at the plot and tell me what onions cost in January. You cannot. And it is worse than not remembering. The dots came off the table sorted, so the order they arrived in is not stored anywhere in the picture. The north town has twelve different prices. There are four hundred and seventy nine million different ways those twelve months could have been arranged to give exactly the picture you are looking at.

4:09The plot cannot choose between them, because it never knew. January is not hidden in there somewhere. January is gone, and if you want it back you go to the table. So keep the table. But look at what the picture handed you in exchange. The two ends, immediately: cheapest and dearest, without hunting. The gap between them, as an actual distance your eye measures rather than a subtraction you perform.

4:40Where the values pile up, and where they thin out. And any value sitting on its own, a long way from the rest, which is the single hardest thing to notice in a column of figures. None of that is in the table. All of it is one glance away here. That is the trade, and every display you will ever meet is making some version of it. The trade pays best when you put two sets on the same line.

5:10The north town above, the south below, same scale, same start, same end. Different shapes, and different colours, so neither of them has to be labelled twice. Now look at the left end. Below twenty rupees, the south has two months and the north has none. And look at how far each one reaches: thirty five rupees end to end against forty three. The south is the more spread out of the two. Clearly, and without measuring anything.

5:41But the south's total was the smaller one. Four hundred and fifty against four hundred and fifty eight. So being spread out and being expensive are simply different things, and the picture separates them where the table did not. Now put the averages back on, because the picture has room for them. A solid line for the mean. The north town's is thirty eight and a bit, the south's is thirty seven and a half.

6:12So the mean says the north town is dearer. Now a dashed line for the middle value. The one with half the months below it and half above. For the north that is thirty seven. For the south, thirty eight and a half. So the middle value says the south town is dearer. Same twenty four numbers. Same picture. Two respectable summaries, pointing in opposite directions. And the picture shows you why, which is something no pair of numbers could do.

6:46Here is the same effect, made obvious. Two families, and everybody's height. The first family: six people, bunched between a hundred and fifty five and a hundred and seventy three. The second: five people, four of them tall — but one down at a hundred and eighteen. Put the means on. A hundred and sixty four for the first family, a hundred and sixty for the second. By the mean, the first family is taller.

7:17Now the middle values. A hundred and sixty four and a half. And a hundred and seventy. By the middle value, the second family is taller, and it is not close. One reading, far out on its own, drags the mean down by more than ten centimetres and moves the middle value not at all. On the plot you can point at the reading that did it. One more pair, and this one is about people rather than prices.

7:48Two groups of children are asked to say when a minute has passed, without counting. Their guesses go on one shared line, in seconds. Both groups average under sixty, so both stopped early. That is the first thing the shared scale gives you. But the shapes are not alike. One group is strung out to the left with a long thin tail of very early stops. The other is packed tightly around the middle, with a tall stack right by sixty.

8:19Two groups, similar averages, completely different pictures — and if all you had been given was the averages you would never have known. So here is the habit worth taking away. Every way of showing data throws something away. That is not a defect, it is the entire mechanism. A total throws away everything except the size. An average throws away every individual case. This plot throws away the order, and buys the shape with it.

8:54So whenever you meet a chart, ask the two questions in that order. What can I see here that I could not see in the raw numbers? And what could I see in the raw numbers that has vanished? You have now met a display where the mean and the middle value disagree, twice, on two completely different data sets. Which means the next question is no longer how to draw the picture. It is which of those two numbers deserves to be believed.