Pictographs, and why the scale has to be stated

0:00Here is a picture you can read before anybody explains it to you. Five rows of little figures. One row for each way of getting to school — private car, public bus, school bus, cycle, walking. And you already know which row is the popular one, because it is the long one. That is the whole appeal of a pictograph. But now let me ask the question that this entire topic turns on. How many students is one of those figures?

0:32And you cannot tell. Nothing inside the drawing answers that. In this one, the answer is printed beside it. One figure stands for one student. That statement is called the key, and with it the rows become numbers. School bus, eleven figures, eleven students. Walking, seven. Public bus, five. Private car, four. Cycle, three. Notice what just happened. The drawing did not change at all. A sentence outside the drawing turned it into data.

1:08And that sentence is the simplest possible one — one for one — which is exactly why it is easy to forget it is there. Now, some questions this picture answers before you have counted anything. Which way do the most students travel? The school bus, and you knew that from across the room, because its row is the longest. Which the fewest? The cycle. Shortest row. You did not do arithmetic there. You compared lengths, which your eye does instantly and reliably.

1:44That is the real argument for a pictograph. Not that it is prettier than a table — that the comparison is done by looking. Now watch what happens when we change the key. A survey asked children in a school how often they had slept nine hours or more. Always, sometimes, or never. And here the key says one symbol stands for ten children. The always row has five symbols. Five symbols, ten children each. Fifty children.

2:18The never row has four symbols. Forty children. Five symbols meant five students on the last picture. Here five symbols mean fifty. Same drawing, different sentence beside it, ten times the number. So the rule to build is: read the key first, every single time. Before you count anything. That leaves the middle row, and it is the interesting one. Sometimes has two whole symbols, and then half of one.

2:51The book calls it a half picture, and it means exactly what it looks like. Half a symbol is half of what the key says. The key here is ten. So a half symbol is five children. Two whole symbols is twenty. Plus the half, five. Twenty-five. And a half symbol does not mean one more. It means half of the key — whatever the key happens to be. So the half symbol doubles what your key can draw. With a key of ten you reach twenty-five and thirty-five too, not just the multiples of ten.

3:28Right. Now you draw one, and drawing is where the thinking is. A teacher records absences for eight classes. Three, five, four, two, none, one, five, seven. One symbol per student. So Class One gets three symbols, Class Two gets five, and you work down the table. And then you reach Class Five, where nobody was absent, and you have to decide what to do. The answer is that you draw nothing. Leave the row empty and move on.

4:01An empty row is not a mistake and not a gap in your drawing. It is a reading. It says zero, and it says it clearly. If you skip that row instead, the pictograph now has seven rows and lies about how many classes there are. Now the same eight classes, but this time the students who were present. Thirty, thirty-five, twenty, twenty-five, thirty, twenty-five, thirty, twenty. One symbol per student, same as before. Class One needs thirty symbols.

4:36And you are already in trouble, because thirty symbols does not fit across a page, and you have seven more rows to go. Altogether that is two hundred and fifteen symbols. Somebody has to draw all of them, and somebody else has to count them. Which defeats the point. The whole reason for a pictograph was reading it at a glance. So the method has not failed. The key has. One for one was the wrong choice for this data.

5:09Two students fix it in two different ways, and comparing them is the heart of this topic. Jarina chooses one symbol for five students. Sangita chooses one symbol for ten. Same eight classes, same numbers, drawn twice. Jarina's rows come out six, seven, four, five, six, five, six, four. Forty-three symbols altogether, and every one of them whole. Sangita's rows come out three, three and a half, two, two and a half, three, two and a half, three, two. Twenty-one and a half symbols. Half the drawing.

5:47Both are correct. Both are readable. They look nothing like each other. And here is the trap that catches people. Jarina's Class Two row is seven symbols long. Sangita's is three and a half. They represent the same thirty-five students. So a longer row means more only when the two rows share a key. Across pictographs, row length tells you nothing at all. Sangita's key is smaller to draw. So why would anyone use Jarina's?

6:22Because of what the bigger key costs, and the book asks exactly the right question to expose it. What if a class had thirty-three students present? Or twenty-seven? With a key of ten, thirty-three is three whole symbols and three students left over. Three is not ten and it is not five, so there is no piece of a symbol that means three. Twenty-seven is two whole symbols and seven left over. Same problem.

6:53And you must not round them. Drawing three symbols for twenty-seven puts thirty on the page, and the picture now says something the data does not. That is the price. A key of ten, with halves, draws multiples of five and nothing else. Jarina's key of five draws multiples of two and a half — a finer net, and a longer drawing. Bigger key, less to draw, fewer values you can be honest about. That is the trade, every time.

7:27So here is what has to be printed beside every pictograph you ever draw. One symbol, an equals sign, and a number. That is the key, and it is not optional. Also what the symbols are counting, and what the rows are — the categories. Leave the key off and you have not drawn a slightly incomplete pictograph. You have drawn a picture with no number in it whatsoever. The rows still have lengths, so a reader will guess a key, and the natural guess is one for one.

8:01So a missing key does not produce confusion. It produces a confident wrong answer, which is far worse. Let us read one we did not draw. Books borrowed from a library, one row per day, one symbol for one book. Monday five, Tuesday four, Wednesday two, Thursday none at all, Friday five, Saturday eight. The busiest day is Saturday with eight. The quietest is Thursday, with an empty row — zero books.

8:34And the week's total? Add them. Five, four, two, nothing, five, eight. Twenty-four books. Notice the total took real work, even though the comparison did not. Six little sums, and no picture helped. Which is worth saying plainly. A pictograph is built for comparing, not for totalling. It answers which is biggest instantly and how many altogether slowly. Last thing, and it is the skill everything else was for. Choosing your own key.

9:09Six people buy kites. Two hundred and fifty, three hundred, one hundred, four hundred and fifty, two hundred and fifty, seven hundred. At a hundred kites a symbol, three of those six land on a half symbol, and three do not. That is workable. Now a harder one, where nobody hands you the key. Dogs counted in six villages — eighteen, thirty-six, twelve, forty-eight, eighteen, twenty-four. You want a key that draws every one of those with whole symbols. So you want a number that divides all six.

9:48One works, and two, and three, and six. And six is the biggest of them. Take six and the rows become three, six, two, eight, three, four. Twenty-six symbols for a hundred and fifty-six dogs, and not a half symbol anywhere. That is the move. Look at your data, find what divides all of it, take the largest such number. The key comes out of the numbers. Next time, we take these rows off the picture and put them on an axis — and the pictograph becomes a bar graph.