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Bar graphs

Statistics and data · level 2, lesson 3 of 6

A question to start

Here are two ways of showing the same survey of 20 children's favourite pets.

Pet Number of children
dog 7
cat 5
fish 3
no pet 4
rabbit 1
Our favourite pets 0 1 2 3 4 5 6 7 8 dog cat fish no pet rabbit number of children

Which one lets you see the most popular pet faster? Which one lets you see that "no pet" beat "fish"? Most people find the picture faster. Your eyes compare heights in a blink, without reading a single number.

That picture is a bar graph. Bar graphs are probably the most used graph in the world: in newspapers, in science, in shops and in schools. In this lesson you learn every part of one, how to choose its numbers, and how to read bars that stop between the lines.

The idea

What a bar graph is

A bar graph shows each count as a bar: a long rectangle. The bigger the count, the longer the bar. It is the picture graph from level 1 grown up: instead of a row of stickers, one solid bar, and instead of counting stickers, you read the length from a ruler drawn on the side.

The parts of a bar graph

Look at the pet bar graph above. It has:

  1. A title at the top: what the graph is about ("Our favourite pets").
  2. Two axes (say "ak-seez"; one of them is an axis). These are the two lines that make an L shape: one along the bottom and one up the side.
  3. Labels for the bars along the bottom axis: dog, cat, fish, no pet, rabbit.
  4. A scale up the side axis: the numbers 0, 1, 2, 3, ... that work like a ruler, telling you how tall each bar is.
  5. An axis label that says what the scale counts ("number of children").
  6. Bars of the same width, with the same gap between them.

Every part has a job. Without the title you would not know what was counted. Without the scale you could see which bar is tallest but not how many. Without the axis label you would not know whether the numbers are children, votes or days.

Every bar starts at zero

The bottom of every bar sits on 0. This matters a lot. A bar graph works because a bar twice as long means twice as many. That is only true if the bars all start at zero.

Watch what happens if someone starts the scale at 3 instead of 0.

Starts at 3: unfair! 3 4 5 6 7 8 dog cat fish no pet rabbit number of children

Now the dog bar (7) looks four times as tall as the no pet bar (4), even though 7 is not even twice 4. The fish and rabbit bars have vanished, as if nobody chose them! The numbers are the same, but the picture lies. A fair bar graph always starts at 0.

Choosing a scale

The scale must count up in equal steps. It can count by 1s, by 2s, by 5s or by 10s, but the steps must all be the same size and the same distance apart, like the marks on a ruler.

How do you choose? Look at the biggest number in your data.

  • If the biggest number is small, like 7, count by 1s. The scale goes 0, 1, 2, ..., 8.
  • If the biggest number is around 10 to 20, count by 2s: 0, 2, 4, 6, ..., 20.
  • If the biggest number is bigger, like 35, count by 5s: 0, 5, 10, ..., 35 or 40.
  • For bigger numbers still, count by 10s.

Choose a scale that makes the graph fit nicely on the page: not so small that the bars are tiny, and not so big that you need a hundred lines. The top of the scale should be at least as big as the biggest number, so the tallest bar fits.

Bars that stop between lines

When the scale counts by 2s, the lines are at 0, 2, 4, 6, and so on. Where does a bar for 5 stop?

5 is between 4 and 6, exactly in the middle. So the bar stops halfway between the 4 line and the 6 line.

Here is the season survey from lesson 1, with a scale that counts by 2s. This time the bars go across (sideways).

Our favourite season 0 2 4 6 8 10 12 winter spring summer fall number of children
  • Summer stops halfway between 10 and 12: that is 11.
  • Winter stops exactly on the 6 line: 6.
  • Spring stops halfway between 4 and 6: 5.
  • Fall stops on the 4 line: 4.

Up or across

Bars can stand up (vertical bars, like the pets) or lie across (horizontal bars, like the seasons). Both are correct, and all the rules are the same. Horizontal bars are handy when the labels are long words, because the words fit nicely at the left.

Order of the bars

For choices like pets or seasons, you can put the bars in any order. Some people put them in the order of the survey. Others put them from tallest to shortest, which makes the most popular and least popular jump out. For seasons, the natural order of the year (winter, spring, summer, fall) is also a good choice. Choose an order and say why.

Worked examples

Example 1. Drawing a bar graph. Draw a bar graph of the pet table.

  1. Title: "Our favourite pets".
  2. Axes: draw a line across the bottom and a line up the left side.
  3. Labels: write dog, cat, fish, no pet, rabbit along the bottom, evenly spaced.
  4. Scale: the biggest number is 7, so count by 1s, from 0 to at least 7. Mark 0 at the bottom line, and the other numbers at equal steps up the side. Label the side axis "number of children".
  5. Bars: above dog, draw a bar up to the 7 line. Cat to 5, fish to 3, no pet to 4, rabbit to 1. Same width, same gaps.
  6. Check: do the bar numbers add up to 20, the number of children? 7 and 5 and 3 and 4 and 1 make 20. Yes.

That is the graph at the top of this page.

Example 2. Reading a scale of 2. Use the season bar graph.

  1. Which season is the favourite? Summer, the longest bar.
  2. How many chose summer? Its bar stops halfway between 10 and 12, so 11.
  3. How many more chose summer than winter? 11 and 6: from 6 count up to 11. 5 more.
  4. How many children answered? 6 and 5 and 11 and 4 make 26.

Example 3. Choosing a scale of 5. A whole school was asked "How do you get to school?" The answers: walk 35, car 20, bus 25, bike 10. Draw a bar graph.

  1. The biggest number is 35. Counting by 1s would need 35 steps, far too many. By 2s would need 18 steps. By 5s needs only 7 steps: 0, 5, 10, 15, 20, 25, 30, 35. That fits well.
  2. Every number in the data is a multiple of 5, so every bar stops exactly on a line.
How we get to school 0 5 10 15 20 25 30 35 40 walk car bus bike number of students

The graph shows straight away that walking is most common and biking least. How many students were asked? 35 and 20 and 25 and 10 make 90.

Try it

  1. Name the five parts every bar graph needs (besides the bars).
  2. In the pet bar graph, how many more children chose cat than fish?
  3. A bar on a scale that counts by 2s stops halfway between 8 and 10. What number is it?
  4. A bar on a scale that counts by 10s stops halfway between 20 and 30. What number is it?
  5. Choose a scale (by 1s, 2s, 5s or 10s) for each data set, and say where the top of the scale should be: (a) 3, 6, 2, 5; (b) 14, 8, 18, 11; (c) 40, 25, 55, 30.
  6. Ella's scale goes 0, 1, 2, 5, 10, 20, with the numbers spaced evenly. What is wrong with it?
  7. Max drew a bar graph of the season data, but the bottom of his scale is 4 instead of 0. Which bar disappears? Why is his graph unfair?
  8. In the travel graph, how many fewer students come by car than walk? How many students do not walk?
Answers
  1. A title, the two axes, labels for the bars, a scale (with equal steps, starting at 0), and an axis label that says what the scale counts.
  2. Cat 5, fish 3: from 3 up to 5 is 2 more.
  3. 9, halfway between 8 and 10.
  4. 25, halfway between 20 and 30. (Halfway is 5 more than 20 and 5 less than 30.)
  5. (a) By 1s, top at 6 (or a little more). (b) By 2s, top at 18 or 20. (c) By 5s or 10s; by 10s the top would be 60, by 5s the top would be 55.
  6. The steps are not equal. From 2 to 5 is 3, from 5 to 10 is 5, from 10 to 20 is 10, but they are drawn the same distance apart. Then bar lengths do not match their numbers: a bar for 20 would look only a little taller than a bar for 10.
  7. Fall (4) disappears completely, as if nobody chose fall. Spring (5) becomes a tiny sliver, and summer (11) looks seven times as long as spring, when 11 is only about twice 5. Bars must start at 0 so their lengths can be compared fairly.
  8. Walk 35, car 20: from 20 up to 35 is 15 fewer. Not walking: 20 and 25 and 10 make 55.

Common mistake

Starting the scale somewhere other than zero. Then small bars vanish and differences look much bigger than they are. In a bar graph, every bar starts at 0.

Uneven steps on the scale. The numbers on the scale must go up by the same amount each step, like a ruler: 0, 2, 4, 6, not 0, 1, 2, 5, 10.

Where it leads