Class 6 Mathematics Chapter 4: Data Handling and Presentation

Meera counts the bottles

Meera is the class monitor of 6B. On Monday her teacher asks her a plain question: how many children bring a water bottle to school? Meera looks around the room and guesses about twenty. Her friend Arun guesses thirty. Neither of them can prove anything. The only way to settle it is to go round the class and find out. That going round, and the writing down that comes with it, is where this chapter begins.

What data is

Anything you write down about the world can become data. A list of favourite fruits is data. The weight of every child in the class is data. So are the number of buses that pass your gate in an hour, and the colour of every door on your street. Facts, counts, measurements and observations all count, as long as they tell you something about the things you were looking at.

Some questions need data, and some do not

Arun asks what the capital of India is. Meera does not go round the class for that. The answer is already settled, so she looks it up. Now Arun asks which snack the class would choose for a picnic. Nobody knows that yet, and no book holds it. Only the class can say. A question that is already answered somewhere needs looking up. A question about this group, right now, needs collecting.

Going round the class

Meera decides to ask every child which fruit they like best. She writes the name of each child and the fruit they say. Thirty children, thirty pairs. It takes a whole break time. At the end she has filled two pages and she is pleased with herself. Then she looks at the two pages and her pleasure fades, because she still cannot answer her own question.

The list that answers nothing

The trouble is that the fruits are scattered. Guava is on line two, then line nine, then line seventeen. To count the guavas Meera must read every line and hold a number in her head, and she loses count twice. The data is all there and none of it is usable. Collecting was only the first half of the job. The second half is putting it in order.

Tally marks keep the count for you

There is an old trick for counting things as they arrive. Write the names down the side. Every time somebody says a fruit, put one short stroke beside it. You never have to remember a number, because the strokes remember it for you. These strokes are called tally marks, and they are the neatest way to count a group of answers that come at you one after another.

Why the strokes come in bundles of five

Four strokes standing side by side are easy to read. Nine are not. So on the fifth stroke you draw a line slanting across the four you already have, closing the bundle. Now the marks read as bundles, and you count them in fives: one bundle, two bundles, and the loose strokes at the end. Eleven marks look like two closed bundles and one stroke left over.

Frequency is just how many times

When Meera counts the strokes beside guava she gets seven. That seven has a name. The frequency of a value is the number of times it turns up in the data. Guava has a frequency of seven, banana eleven, sitaphal three. The word sounds grand and means something very ordinary: how often. Every question Meera wanted to answer is a question about frequencies.

A frequency table

Now Meera rules three columns. In the first she writes each fruit. In the second she draws the tally marks. In the third she writes the frequency as a number. This is a frequency table, and it does what two pages of names could not: the whole class fits in five rows, and the most popular fruit is the row with the longest run of strokes.

The check that catches a slip

Meera adds her frequencies: seven and four and eleven and five and three. That comes to thirty, which is exactly how many children she asked. This addition is worth doing every single time. If the total had come to twenty nine, somebody was missed or a stroke was lost. If it had come to thirty one, somebody was counted twice. One sum, and the mistake shows itself.

Another way to organise: put them in order

Fruits have no natural order, but numbers do. Meera's teacher asks a second question: how many storybooks did each child read in the holidays? The answers come back as twenty scattered numbers. Meera writes them again from smallest to largest. Suddenly all the ones sit together, then the twos, then the threes. Nothing has been added or removed. Only the arrangement has changed.

What the ordered list hands you

With the numbers in order, the smallest value is sitting at the front and the largest at the back, so you can read them straight off. Counting how many children read three books means counting one run of threes instead of hunting through the whole list. Ordering does not answer the question by itself. It puts the answer where you can see it.

Choosing how to organise

So there are two useful moves. Tally marks and a frequency table suit answers that are names, such as fruits or games or colours. Ordering suits answers that are numbers, such as shoe sizes or books read, and it feeds a frequency table too. Neither move changes the data. Both of them turn a heap of answers into something a person can actually read.

Scenario:

A class surveys how children travel to school. Rohit asks the first ten pupils he meets in the corridor and reports that most come by bus. Their teacher asks where he was standing. By the gate the buses use. The class repeats the survey across every section and finds walking is the commonest. Rohit sees that his ten answers were honest and his sample was not. The class writes down what a survey needs. Everybody in the group must have the same chance of being asked, or the place you stand decides the answer before anybody speaks.

Analysis:

Rohit's ten answers were all true and the conclusion was still wrong. Where he stood selected his sample for him.

Activity:

Survey how your class travels to school, asking everybody. Then ask only ten at one place and compare the two results.

Title:

A tally of the road outside

Safety:

Stay behind the gate or at the window. Do not stand on the road itself.

What You Notice:

The bundles let you keep counting without ever pausing to add up. The frequency that is largest is usually the one you would have guessed, but the smaller ones are often a surprise. Try the same fifteen minutes on a Sunday and the table comes out quite different.

Shepherds once cut notches into a stick to count sheep, which is the same idea as a tally mark carved in wood.

Cricket scorers still keep a running tally by hand as a backup, in case the electronic scoreboard fails.

India counts its whole population in a census, which is data collection on a scale of crores of people.

Drawing tally marks — Four stand up, the fifth lies across — Draw four uprights, then close the bundle with a slanting stroke over them. Every closed bundle you see is worth five.

Checking a frequency table — Add the column, match the crowd — Add every frequency. The total has to equal the number of people or things you collected from, or something went wrong.

Rani puts her week on the wall

Rani keeps a stationery shop beside the bus stand. At the end of every week she writes down how many notebooks she sold. Five weeks in, she has five numbers on a scrap of paper. Her nephew Faiz says nobody will read a scrap of paper. He suggests drawing a little notebook for every ten she sold and pinning the whole lot on the wall. Rani tries it, and customers start pointing at the wall.

What a pictograph is

A pictograph shows data using small pictures instead of written numbers. Each row or column stands for one category, and the pictures are lined up along it. The longer the line of pictures, the larger the number it is standing for. You can compare two rows without reading a single digit, which is why pictographs turn up in newspapers and on the walls of shops.

Reading one at a glance

Look at Rani's wall and you know at once that week four was her best week and week three her worst. You did not count anything. You compared the lengths of two lines of pictures, the way you would compare two queues. That is the whole point of a pictograph: the first answer arrives before any arithmetic starts.

The key does the real work

A picture of a notebook is not worth anything until somebody says what it is worth. That statement is called the key, and it is written beside the pictograph: one notebook picture stands for ten notebooks sold. Without the key the drawing is decoration. With it, every row can be turned back into a number. Never draw a pictograph without writing its key.

When one symbol means one thing

Sometimes the key is the simplest one possible. If eight children in a class wear spectacles, you can draw eight little pairs of spectacles and write that one picture stands for one child. This is easy to read and easy to check, because the pictures and the children match one for one. It works well while the numbers stay small.

When the numbers get big

Now imagine drawing one picture for each of Rani's two hundred and fifty notebooks. Your arm would ache and the wall would not hold them. Worse, nobody could count them. Once the numbers grow past a couple of dozen, a key of one picture for one thing stops helping and starts getting in the way.

Choosing a larger key

So you let one picture stand for more. Rani chose ten. Now forty notebooks is four pictures, sixty is six, and seventy is seven. The wall stays small and the rows stay countable. The bigger the numbers in your data, the bigger the key you should pick, so that the longest row is still short enough to take in.

Half a symbol for half the value

Rani sold twenty five notebooks in week three, and twenty five is not a whole number of tens. So she draws two whole notebooks and then half of one. The half picture is worth half the key, which is five. Two tens and a five make twenty five, and the row is honest. A part symbol has to look like the part it stands for, so a half must be clearly half.

Reading a row back into a number

Here is the habit to build. Count the whole symbols. Multiply by the key. Add the value of any part symbol. Week five shows five whole notebooks and a half, so that is five tens, which is fifty, and then five more, giving fifty five. If you cannot get the original number back out of the row, the row was drawn wrong.

The same number, many keys

Sixty notebooks can be drawn as twelve pictures with a key of five, or six pictures with a key of ten, or three pictures with a key of twenty. All three rows are correct, because each one carries its own key. They simply cost different amounts of drawing. This is worth seeing once, because it shows that the key is a choice and not a fact about the data.

Every symbol the same size

A pictograph compares rows by counting pictures, so the pictures have to be identical. If one week's notebooks are drawn larger than another's, the eye reads the bigger ones as more, and they are not. Same drawing, same size, every time. Only the number of them may change.

When a key does not fit

Suppose a key of ten and a value of thirty three. Three whole pictures leave three over, and three is not half of ten. You cannot draw a neat part symbol for it. This is the real limit of pictographs, and there are two honest ways out: choose a key that divides your numbers, or move to a bar graph, which can show any value at all.

What pictographs are good for

They are friendly, quick to read, and need no scale along an edge. They suit small sets of data with tidy numbers, and they are excellent on a wall or a poster. They are poor at awkward values and slow to draw for large ones. Knowing both halves of that is what lets you choose the right picture for your data.

Scenario:

A class draws a pictograph of fruit sold in a week. Nita uses one apple symbol for every two fruits sold. Imran uses one symbol for every ten. Their charts look very different, and Imran's is much shorter. Imran says Nita has exaggerated. Their teacher asks the class to read the actual numbers off both. They agree exactly. The class sees that neither chart is wrong, because the key is doing the work. Nita writes the warning down. A pictograph cannot be compared with another until you have read its key, since the same bar length can mean two or ten.

Analysis:

Two charts of one week's sales, both correct, looking nothing alike. The key is the part that makes a symbol mean something.

Activity:

Draw one week of data as a pictograph twice, with different keys. Ask a friend to read both and check they agree.

Ancient traders drew small pictures of jars and animals on clay to record a load, long before written numerals existed.

Airport and railway signs are picture based for the same reason a pictograph is, because a picture crosses languages.

Weather pages still use one cloud or one sun picture for each day, which is a pictograph with a key of one.

Reading a pictograph row — Count, times, add — Count the whole symbols, multiply by the key, then add whatever a part symbol is worth.

Never forget the key — No key, no meaning — A row of pictures with no key beside it cannot be turned back into a number by anybody.

The picture on the back page

Tanvi finds a graph on the back page of the newspaper. Coloured columns of different heights stand in a row, with words underneath them and numbers up the side. She does not read the article, but within a few seconds she can say which column is tallest and roughly how much taller it is than its neighbour. A pictograph would have needed hundreds of little drawings for the same numbers. This is a bar graph.

What a bar graph is

A bar graph turns each category into one bar. The bar's length or height stands for the number belonging to that category. Bars stand side by side with spaces between them, and a scale runs up one side so that any bar can be turned back into a number. It is the display of choice once a pictograph would need too many symbols.

Every bar the same width

The bars in a bar graph are all drawn the same width. This is not tidiness, it is honesty. If one bar were fatter than the rest, it would look like more even when its height said otherwise, and the reader would be comparing two things at once without being told. Only the height is allowed to carry meaning.

Equal gaps between the bars

The bars also sit with equal spaces between them. The spaces say that the categories are separate things standing on their own, like guava and banana, and that no category is closer to another. Uneven gaps make some bars look crowded together and others look important on their own, which the data never said.

The scale, and where it starts

Up the side of the graph runs the scale, marked in equal steps. The scale states what one step is worth: one bird, ten runs, six hundred rupees. Two rules hold it together. The steps must all be the same size, and the markings must begin at zero. A scale that starts anywhere else makes short bars vanish and long ones look enormous.

Reading a value off a bar

Put your finger on the top of the bar and slide it straight across to the scale. The rung it meets is the value. On Tanvi's pond graph, Wednesday's bar reaches the rung marked twelve, so twelve birds were counted. Notice what you did not do: you did not measure the bar with a ruler and you did not count pixels. You read it against the scale.

Comparing two bars

Once the scale is fixed, the bars compare themselves. Monday's bar reaches eight and Thursday's reaches three, so Monday had five more birds, and the difference is visible as the gap between their tops. Because every bar is the same width and every step of the scale is the same size, that comparison by eye is trustworthy.

Lying down or standing up

Bars can grow upward from a line along the bottom, or rightward from a line down the left. Both are bar graphs and both are correct. A bar graph whose bars stand up is also called a column graph. The direction changes nothing about the numbers. It only changes which way the reader tilts their head.

Starting to draw: two lines

Now to make one. Begin with two straight lines meeting at a corner, one running across and one running up. The corner where they meet is the zero of the graph. Everything that follows hangs on those two lines, so draw them with a ruler and leave plenty of room above and to the right.

Where the categories go

Along the line that runs across, write the name of each category, spaced evenly. These are the places the bars will rise from. Along the line that runs up, the numbers will go. Tanvi writes Mon, Tue, Wed, Thu and Fri across the bottom, each the same distance from the next, and leaves the side clear for her scale.

Choosing the scale: one unit, one bird

Her largest count is twelve birds. Twelve steps fit easily on a page, so she takes one unit of length to stand for one bird and marks her side line from zero to twelve. Now Monday's bar rises eight units, Tuesday's five, Wednesday's twelve. Each bar is exactly as many units tall as the number it stands for, which makes this the easiest scale of all to draw and to read.

When one for one stops working

The school cricket team scored ninety runs in one match. At one unit for one run, that bar would be ninety units tall. The page is not that long, and nobody wants to rule ninety markings. The very thing that made the bird graph easy has become the thing making this graph impossible.

A bigger step for bigger numbers

So let one unit stand for ten runs instead. Ninety runs is now nine units, seventy is seven, twenty is two. The tallest bar is nine units and the whole graph fits comfortably. The rule behind the choice is simple: pick a step that keeps the tallest bar on the page while still letting every value be marked exactly.

What the scale must always say

Whichever step you choose, write it down beside the graph, as in one unit length equals ten runs. A reader who cannot see the step cannot turn a bar back into a number, and a bar graph without a stated scale is as useless as a pictograph without a key.

Scenario:

A class is shown a bar graph of two cricket scores, 52 and 56. On the chart one bar looks almost twice the other. Arjun says one team scored twice as much. Their teacher asks the class to read the numbers up the side. The scale does not start at zero; it starts at 50. The bars are showing 2 and 6 above the baseline rather than 52 and 56. The class redraws it from zero and the two bars become nearly the same height. Arjun sees that the graph was accurate and the impression was not, because a cut baseline exaggerates every difference on it.

Analysis:

Every number on the chart was correct. Starting the scale at 50 made a four-run gap look like a doubling.

Activity:

Draw two close numbers as bars twice, once from zero and once from just below them. Write which gives a fairer impression.

Title:

Five minutes at your window

What You Notice:

Counting is harder than it sounds, because birds move. Writing the numbers down first stops you from bending the graph towards what you expected. When the bars are up, the largest and smallest mornings are obvious without reading a single number, and that is the whole reason for drawing it.

The bar graph was invented by a Scottish writer in the seventeen eighties, who wanted to show trade figures to people who hated tables.

Election results on television are almost always shown as bars, because a viewer has only seconds to take them in.

A graph with upright bars is called a column graph, after the pillars that hold up a roof.

The three rules of a fair bar graph — Same width, same gap, start at zero — Check those three before you trust any bar graph you meet, including your own.

Reading a bar — Top across to the scale — Slide across from the top of the bar until you meet a rung. That rung is the value.

The money from the school fete

The fete raised money at five stalls. Books took one thousand eight hundred rupees, food three thousand six hundred, games one thousand two hundred, plants six hundred and crafts two thousand four hundred. Sunita has to put this on a poster by Monday. She starts to rule a scale of one unit to one rupee, gets to twenty, and stops. At that rate the food stall would need a bar three thousand six hundred units tall.

When the numbers are far too big

This is the same difficulty as the cricket scores, only much worse. A step of one rupee is useless here. The bars would run off the poster, off the wall and out of the room. Whenever the largest value is in the hundreds or thousands, the first decision is not where to draw but what one unit is going to be worth.

Choosing a step that divides

Sunita looks at her five numbers and asks what goes into all of them. Six hundred does. One thousand eight hundred is three sixes of a hundred, three thousand six hundred is six of them, and the smallest stall, plants, is exactly one. So she takes one unit of length to stand for six hundred rupees. A step that divides every value means no bar will have to be guessed at.

The rule that gives every height

Here is the rule that does all the work. Take a value and share it into steps the size of the scale. The number of steps is how far that bar rises. Books is one thousand eight hundred, which holds three six hundreds, so its bar is three units. Food is three thousand six hundred, which holds six of them, so its bar is six units. The sharing is done once for each stall, before any bar is drawn.

Writing the heights down first

Sunita makes a small table of three columns: the stall, the division, and the answer in units. Books three, food six, games two, plants one, crafts four. It takes two minutes and it saves the poster. Every height is now a small whole number she can count out on squared paper, and if she has made an arithmetic slip, she will find it here rather than halfway through the drawing.

Drawing from the table

Now the drawing is mechanical. Rule the two lines. Write the five stall names evenly along the bottom. Mark the side from zero in equal steps, labelling them six hundred, one thousand two hundred, one thousand eight hundred and so on. Then raise each bar to the number of units in the table, all the same width, with the same gap between each pair.

Checking by reading it back

When the bars are up, Sunita checks her own work the way a stranger would. She takes the top of the food bar across to the scale and reads three thousand six hundred. She takes the plant bar across and reads six hundred. Both match the numbers she started with. A graph you cannot read back to your own data is a graph with a mistake in it.

A graph with a piece missing

Her teacher gives the class a different puzzle: a bar graph of letters posted in five boxes, with the side markings rubbed out. Two bars carry their values above them. The box at the market is labelled eighteen and stands three units tall. The box at the station is labelled twenty four and stands four units tall.

Working out the scale from a labelled bar

If three units are worth eighteen letters, then one unit is worth eighteen divided by three, which is six. The station bar agrees: four units at six letters each is twenty four. Two bars that agree is enough to trust the scale. Now the markings can be written back in as six, twelve, eighteen, twenty four and thirty.

Filling in what is missing

The school box stands two units tall, so it carries two sixes, which is twelve letters. The hospital box stands five units, so five sixes, which is thirty. And a bar that has to show six letters is drawn one unit tall. Once the scale is known, every missing piece follows from it, in both directions.

Looking for mistakes in a finished graph

The last skill is the sharpest one. Given somebody else's graph, can you find what is wrong with it? There are three faults to hunt for, and they are the three rules turned upside down: bars of different widths, gaps of different sizes, and a scale that does not start at zero.

Why the cut scale is the worst fault

Unequal widths and gaps look untidy, and untidy things get questioned. A scale that starts partway up looks perfectly neat, and that is what makes it dangerous. Cutting the same slice from the bottom of every bar leaves the tall ones barely touched and the short ones almost gone, so one bar can seem four times another when the numbers are nowhere near that.

Putting it right

To repair such a graph, redraw the side line from zero in equal steps and put every bar back to its true number of units. Give all the bars one width and one spacing. The picture that comes out will look less dramatic than the one that went in. That loss of drama is the mistake being removed.

What the graph gives you that the table did not

Sunita still has her table of five numbers, and it is perfectly correct. But nobody walking past a poster reads a table. The bar graph makes the food stall's lead visible from across the corridor, shows the plant stall as the shortest, and invites the next question: what should we do differently next year?

Scenario:

A class draws a bar graph of the books each section borrowed in a month. The numbers run from 8 to 46. Meena chooses one centimetre for every book and runs off the bottom of her page. Rohit chooses one centimetre for every ten and his bars are too short to compare. Their teacher asks the class what a scale has to do. It must fit the largest value on the page and still separate the closest two. The class tries one centimetre for every five, which puts 46 at just over nine centimetres. Both conditions are met and the graph becomes readable.

Analysis:

Two sensible-looking choices, both unusable. A scale has to fit the biggest value and still show the smallest difference.

Activity:

Collect data with a range like 8 to 46. Try three scales and write which one fits the page and still shows the differences.

Title:

Draw the fete poster

What You Notice:

The drawing takes far less time than choosing the scale did, which is usually the way round. When you read the bars back, any slip in the division shows up at once, because the bar gives you a number you never wrote down. Try the whole thing again with a step of twelve hundred and you will see the graph get shorter and harder to read.

Newspapers sometimes cut the bottom off a scale on purpose, because a dramatic graph sells more copies than a fair one.

Squared paper was made popular in schools partly because it makes equal bar widths and equal gaps almost automatic.

Scientists are expected to show the zero on a chart, and a paper can be sent back for review if the axis is cut.

Finding the height of a bar — Value over scale — Divide the value by what one unit is worth. The answer is how many units tall to draw the bar.

Checking someone else's graph — Widths, gaps, zero — Look at the three in that order. Two are visible at a glance, and the third needs you to find the bottom of the scale.

Ishan enters the poster competition

Ishan has measured five trees in the school compound. The neem is twelve metres, the amla four, the mango ten, the teak sixteen and the ashoka eight. His numbers are correct and his table is neat, and the judges walk straight past it. Next to him, Bhavna has the same sort of data on a large bright graph, and a small crowd has gathered. Ishan learns something that afternoon which has nothing to do with arithmetic.

Fitting the space you have

The first thing a display must do is fit. A graph that runs off the edge of the sheet has failed before anybody reads it, and one squeezed into a corner is too small to see. The tool for this is the scale. A larger step makes the bars shorter, a smaller step makes them taller, and the data is untouched either way. Choose the step that fills the space you have been given.

Making it worth looking at

After it fits, a display should invite a reader in. Clear colours, a real title, labels that can be read from a few steps away, and enough white space that the bars are not fighting each other. None of this changes a single number. All of it changes how many people will stop and look at those numbers.

Heights like to stand up

Ishan's trees are heights, measured upward from the ground. If he draws them as bars growing upward, the picture works the way the world works, and a reader understands it without being told. A bar graph with upright bars has its own name: a column graph, after the pillars that hold up a roof.

Lengths like to lie down

Now think about the walking trails around his town, of four, ten, six, fourteen and eight kilometres. These are lengths along the ground, not heights. Drawn as bars running rightward from a line down the left side, they match what they stand for. Distances between places are almost always shown with horizontal bars for exactly this reason.

The direction is only a kindness

Here is the important part. Either direction is mathematically correct. Ishan could draw his trees as horizontal bars and every value would still be right. Turning the graph does not make it truer, only friendlier, because the reader spends less effort picturing what the bars stand for.

When a graph is dressed up

Sometimes a graph is decorated much further, with drawings, colours and pictures of the thing being measured. A display like that is called an infographic. The aim is to catch an eye in a corridor or on a newspaper page and to hold it long enough for the numbers to land.

Why infographics are tempting

They work. A row of little trees is friendlier than a row of grey rectangles, and more people will stop. That is a real gain, and there is nothing wrong with wanting it. The trouble begins when the decoration starts to carry meaning that the data never gave it.

Bhavna's hills

Bhavna redraws Ishan's trees as triangles, so that they look like hills. Every peak is placed at exactly the right height against her scale. But she also makes the taller triangles wider, because a tall thin spike looked odd to her. The picture is lovely and it is now telling two stories instead of one.

What the extra width claims

A reader looking at her drawing sees the teak hill as wide as it is tall, and the amla as a small narrow point. The eye compares whole shapes, not heights alone. So the teak looks very much more than four times the amla, though four times is all the numbers said. Nobody wrote that claim down. The drawing made it anyway.

Checking a picture against its numbers

There is a simple test for any decorated graph. Pick two shapes, read their values off the scale, and work out how many times one is the other. Then ask whether the picture looks like that many times. Teak is sixteen metres and amla is four, so teak should look four times amla. If the drawing suggests ten times, the drawing is wrong, however beautiful it is.

Same width, one message

The cure is the rule from earlier, now seen from a new angle. Keep every shape the same width, and the only thing that varies is the height. Then the eye has just one thing to compare, and it compares it correctly. Decorate all you like above that rule, and never below it.

Careful both ways

This matters twice over. When you are making a display, you must not mislead your reader, even by accident and even while trying to be helpful. When you are reading somebody else's, you must not be misled. Posters, advertisements and newspaper pages are full of pictures that are pretty first and accurate second.

What Ishan does next year

He keeps his careful measuring and adds a clear title, strong colours and bars that fill the sheet. He draws the trees standing up because they are heights. He keeps every bar the same width. The poster is attractive and it is honest, and those two things turned out not to be enemies after all.

Scenario:

The panchayat of Rampur wants a noticeboard showing how much of the village water supply reaches each of its four wards. Meena, who is in Class 6, is asked to draw it. The readings are given in thousands of litres a day, and they are four thousand, six thousand, two thousand and eight thousand. The secretary suggests drawing each ward as a water pot, with the bigger wards shown as bigger pots all round, because that would look attractive on the board. Meena is not sure. She remembers being told that only one thing may change from shape to shape. She has a large sheet of paper, coloured pencils and one afternoon in which to finish the work.

Analysis:

A display for a village noticeboard must be attractive and honest at once, and keeping every shape to one width is what lets it be both.

Activity:

Draw the same four readings twice, once as bars of one width and once as pots that grow in every direction, and ask two people at home which ward looks worst off in each drawing.

A nurse in the eighteen fifties drew coloured diagrams of army deaths to persuade a government to act, and it worked.

Railway timetables were among the first documents to use a graph rather than a table, so that travellers could read them quickly.

Newspapers employ designers whose whole job is to draw data, because a picture is read far more often than a table.

Choosing the direction of the bars — Tall things stand, long things lie — Heights go up the page and lengths go across it, so the picture matches the thing being measured.

Testing a decorated graph — How many times? Does it look it? — Read two values, work out how many times one is the other, then ask whether the drawing really looks that way.