Drawing a bar graph, and choosing a scale that fits

0:00We have read other people's bar graphs. Now we draw one — and there is exactly one place where you have to think. Start with a table we already built. Five sweets, and how many children asked for each. Ice cream thirteen, chocolate cake nine, cookies seven, doughnuts six, brownies three. That is the input; a picture of it is the output. In between is a four-step procedure, three steps of which are mechanical.

0:32Step one. Draw two lines meeting at a corner, one across and one going up. Now decide what each one carries, and there is a convention here worth following. The categories — the five sweets — go along the bottom. The numbers go up the side. Mark the bottom line into five equal slots, one per sweet, and label them. The upright line gets numbered — and here is the rule the chapter ends on, taken early: the numbering starts at zero.

1:06Not at three because your smallest value is three. At zero. Every time. Step two, and this is the only real decision in the whole procedure. How many students will one unit of length be worth? Look at what you are drawing. The smallest frequency is three and the largest is thirteen. So one unit for one student is comfortable — thirteen units is a fine height, and the axis runs zero to about fourteen.

1:39And notice where that answer came from. Not from a rule, not from a teacher — from the smallest and largest numbers in your own data. That is the whole skill. Everything after this is division. Steps three and four, and they are mechanical. Above each label, draw a bar. Its height is the frequency divided by the scale — and the scale is one, so the height is the frequency.

2:09Ice cream, thirteen units. Chocolate cake, nine. Cookies, seven. Doughnuts, six. Brownies, three. Every bar the same width. Equal gaps between them. And that is a finished bar graph. Four steps, and only one of them needed a decision. Now watch what happens when the data makes that decision hard. A batter's scores across eight innings. Eighty, fifty, ten, a hundred, ninety, nought, ninety, fifty. Try one unit per run and see what goes wrong.

2:45The upright line would have to be marked nought, one, two, three — all the way to a hundred. A hundred marks on one line. You would be ticking all afternoon, and the graph would be taller than the page. But look at the numbers themselves. Every single score is a multiple of ten. The data is telling you what scale it wants. Take one unit of length to be ten runs.

3:13Now redo it. The axis reads nought, ten, twenty, up to a hundred. Eleven marks instead of a hundred and one. And every height is a division. Eighty divided by ten is eight units. Fifty divided by ten, five units. A hundred divided by ten, ten units. Which brings us to the sixth innings, where the batter was out for nought. Zero divided by ten is zero. So the bar has no height at all.

3:46And you keep the slot. Label innings six, draw nothing above it, and move on. Delete the slot and the record is wrong — it would claim seven innings, and hide a duck, which is exactly what a record is for. Here is the example that makes the division visible, because the book prints every one of them. A family's monthly spending. Rent three thousand rupees, food three thousand four hundred, education eight hundred, electricity four hundred, transport six hundred, everything else twelve hundred.

4:26One unit of length is taken to be two hundred rupees. So now just divide. Three thousand over two hundred is fifteen units. Three thousand four hundred over two hundred, seventeen units. Eight hundred, four units. Four hundred, two. Six hundred, three. Twelve hundred, six. Fifteen, seventeen, four, two, three, six. That column is the graph. You have drawn nothing yet, and the drawing is already fixed. Which is the point. Bar heights are not drawn by eye. They are computed, then measured out with a ruler.

5:07Now read your graph back, because two of the questions the book asks are sharper than they look. Which item takes the most? Food. Second? Rent. Those you can see. Now: is the electricity bill about half of what is spent on education? Education is eight hundred. Half of eight hundred is four hundred. Electricity is four hundred. Not about half — exactly half. And the harder one. Is education less than a quarter of what the family spends on food?

5:41Do not eyeball this. Compute it. A quarter of three thousand four hundred is eight hundred and fifty. Education is eight hundred. So yes — but by fifty rupees. Had food been three thousand two hundred, a quarter would be eight hundred and the answer flips. That is why you work the quarter out instead of squinting at two bars. The true answer was fifty rupees away from the false one.

6:11Now a puzzle that runs the procedure backwards. Somebody drew tickets sold at a station, and part of it has been rubbed off the board. The bars are still there. The numbers up the side are gone. So you do not know the scale. But you know two things. The bar for one destination is six units long, and twenty-four tickets were sold there. Six units is twenty-four tickets. So one unit is twenty-four divided by six. Four tickets.

6:44Check it against another bar first. A second destination is five units long, twenty tickets sold. Five fours are twenty. It holds. And now the whole graph is readable again, from one surviving bar and one number. With the scale recovered, two things on that board are wrong. One destination sold twenty-eight tickets. Twenty-eight divided by four is seven units. But its bar is drawn somewhere between four and five.

7:19So that bar is simply wrong. Not a rounding — wrong by nearly half its length. And a fifth destination, which sold sixteen tickets, has no bar at all. Sixteen divided by four is four units. So it should be there, exactly as tall as the sixteen beside it. Two different faults, both found by the same division you would have done to draw the graph. Checking a picture against its table is a real skill, and the book teaches it on purpose by handing you a broken graph.

7:56One more, and it is the best trap in the chapter. A bowler's thirty matches, arranged by wickets taken. No wickets in two matches, one wicket in four, two in six, three in eight, four in three, five in five, six in one, seven in one. First, check the table is complete. Add the match column. Two, four, six, eight, three, five, one, one. Thirty. Good — every match is there.

8:27Now, how many wickets altogether? And somebody in the book says: easy, add the left column. Nought plus one plus two, up to seven. Twenty-eight. That is wrong, and it is worth seeing how wrong. The left column lists wicket values, not matches. Three wickets happened in eight matches, so that row alone gives three times eight — twenty-four. Do every row that way and the real total is ninety wickets. The answer that felt easy was twenty-eight. It is not a bit off. It is under a third of the truth.

9:07So, the rules the chapter closes on, and now you know why each one is there. Every bar the same width — so that width carries no information and only length speaks. Equal gaps between bars — so the eye sees separate categories, not one continuous quantity. And the markings start at zero — because starting at fifteen makes a bar of sixteen look tiny beside one of twenty-eight, which the numbers do not say.

9:39Those three are not style. They are what makes a length recoverable back into a number. And the whole procedure comes down to one line: choose the scale from your smallest and largest values, then every height is frequency divided by scale. Next time, the last question in the chapter — which way should the bars run, and does it change what people see?