Two graphs show the same numbers. Why does one line climb so much more steeply?
Your guess
Pick one before you read on. It is fine to be wrong.
By the end you can
Draw a line graph for data that changes over time
Spot a chart that misleads by not starting at zero
A line graph shows how something changes over time, such as the height of a plant each week. Each point is a measurement. The straight lines join the points so you can see the trend.
Draw it first
Draw a line graph of a growing plant
1 Write the measurements in a table: the week and the height.
2 Mark a point for each week at its height. Join the points with straight lines, from left to right.
Worked example
A plant is measured each week. The heights in cm are 10, 14, 20, 24 and 30. Draw a line graph. How much did the plant grow from week 1 to week 5?
The tallest height is 30 cm. A scale from 0 to 30 in steps of 10 is enough.
Put the weeks 1 to 5 along the bottom. Mark a point for each height. Week 3 is at 20.
Join the points with straight lines, from left to right.
Read the graph. From week 1 to week 5 the plant grew 30 − 10 = 20 cm.
Answer: 20 cm
The line goes up every week, so the plant grows every week.
See it three ways
With things
A bean plant is 10 cm tall in week 1 and 14 cm tall in week 2. Hold a ruler beside it each week and write down the height.
As a picture
In symbols
Week 1: 10 cm. Week 2: 14 cm. It grew 14 − 10 = 4 cm.
Now look at a library. It lends 42, 44, 47 and 50 books on four days. Here are the same numbers on two graphs.
Same numbers. The first scale starts at 0 and the second at 40. The second line climbs five times as far.
In the second graph the part of the scale below 40 is cut away. The line looks as if the library lends far more books each day. But it lends only 8 more on Thursday than on Monday. A bar chart must start its scale at 0, because the height of a bar is its count. A line graph that starts above 0 must say so with a zigzag.
Worked example
A bar chart shows 42 books on Monday and 50 books on Thursday. Its scale starts at 40. How many more books were borrowed on Thursday? Is the chart fair?
Read the bars: Monday is 42 and Thursday is 50.
Take away: 50 − 42 = 8 more books on Thursday.
The scale starts at 40, so every bar is cut. The Thursday bar looks 5 times as tall as the Monday bar. But 50 is not 5 times 42.
A fair chart starts at 0. Then the bars are 42 and 50 tall and look nearly the same.
Answer: 8 more books. The chart is not fair because the scale starts at 40.
The scale starts at 40. Thursday's bar looks 5 times as tall as Monday's, yet it holds only 8 more books.
Sometimes a graph must be cut, because all the numbers are large and close together. An honest graph then draws a zigzag on the axis. The zigzag says: part of the scale is missing.
The zigzag near the bottom of the side axis is a warning. This scale does not start at zero.
Try it
Draw it twice
You need: squared paper or plain paper, a pencil and a ruler.
Take the numbers 42, 44, 47 and 50.
Draw a bar chart with a scale from 0 to 50 in tens.
Draw a second bar chart of the same numbers, with the scale from 40 to 50 in twos.
Hold the two drawings side by side. Which one makes the rise look bigger?
What you will see: The bars on the first chart are nearly the same height. The bars on the second chart climb steeply. The numbers did not change. Only the scale did.
If you cannot try it, this is what happens.
A wrong idea many people have
A steeper line always means a bigger change.
Why it feels right: A steep line looks like a lot of change, and the picture is what we notice first.
The test that breaks it: Look at the scale of each graph. Both line graphs show the same 8 books of change. One starts at 0 and the other at 40.
What is true: Read the numbers on the axis. Where a graph starts and how big each step is decide how steep a line looks.
Key words
line graph
a graph with points joined by lines, for data that changes over time.
scale
the numbers along an axis, with equal steps.
misleading
giving a wrong picture, even if the numbers are right.
Back to your guess
The answer is C: The steeper graph starts its scale above zero, so part of the graph is cut away.
Both graphs show the same numbers. The steep one starts its scale at 40 instead of 0, so small changes look big. Always read the numbers on the axis before you trust a picture.
Check yourself
Try each one before you look. A wrong answer tells you where to go next.
A bar chart has a scale that starts at 40 and has no zigzag. Why can this be misleading?
See the answer
It makes small differences between the bars look big.
The bottom of the scale is cut away. The bars that remain look very different in height, even if the numbers are close.
If you chose B: The numbers do not change. The bars are only cut at the bottom, so they look different. Go to M3.7.5.
If you chose C: It does the opposite. When the bottom is cut away, the gaps between the bars look larger. Go to M3.7.5.
Read the line graph. The scale goes up in steps of 2. How many books were borrowed on Wednesday?
See the answer
47 books
The Wednesday point is halfway between the line for 46 and the line for 48. Halfway is 47.
If you chose A: 46 is the line just below the point. The point is halfway up to the next line, so read between the lines. Go to M2.7.6.
If you chose C: 48 is the line just above the point. The Wednesday point does not reach it. Go to M2.7.6.
Mei measures the temperature each hour through a day. Which graph is best?
See the answer
A line graph
A line graph puts time along the bottom. It joins the points, so you can see the temperature rise and fall.
If you chose A: Class bars show how many values fall in each group. They do not show change through time. Go to M3.7.1.
If you chose B: A pie chart shows how one whole is shared at one moment. Temperature changes over time, so use a graph with time along the bottom. Go to M3.7.2.
Thursday's bar looks 5 times as tall as Monday's. Is Thursday's book count 5 times Monday's?
See the answer
No. Thursday has only 8 more books than Monday.
Monday is 42 and Thursday is 50. The difference is 50 − 42 = 8. The cut scale made it look much bigger.
If you chose B: The scale starts at 40, so only the top of each bar shows. Read the numbers: Monday is 42 and Thursday is 50. Go to M3.7.5.
If you chose C: Thursday's bar is the taller one. It has more books, but not 5 times as many. Go to M3.7.5.
Which sign warns you that the scale of a graph does not start at zero?
See the answer
A zigzag on the axis
A zigzag on the axis means part of the scale is left out. Always check the first number on the axis too.
If you chose A: Gridlines help you read values. They say nothing about where the scale starts. Go to M3.7.5.
If you chose C: Joining points is how a line graph is drawn. It does not warn that the scale is cut. Go to M3.7.5.
A line graph shows Tuesday at 44 books and Thursday at 50 books. How many more books were borrowed on Thursday?
See the answer
6 books
Take away: 50 − 44 = 6.
A graph has a scale from 40 to 50, going up in steps of 2. How many steps are there?
See the answer
5 steps
The scale covers 50 − 40 = 10. Each step is 2, so 10 ÷ 2 = 5 steps.
From an earlier chapter
Review: both groups have a mean of 6. Group X is 5, 6 and 7. Group Y is 2, 6 and 10. How much bigger is the range of Y than the range of X?
See the answer
6
The range of X is 7 − 5 = 2. The range of Y is 10 − 2 = 8. The difference is 8 − 2 = 6.
Keep it
Copy this into your exercise book.
A line graph shows change over time. Join the points with straight lines. Read the scale before you read the picture. A bar chart starts at 0. A cut line graph needs a zigzag.
Draw: Draw the numbers 42, 44, 47 and 50 as a line graph twice. Start one scale at 0 and the other at 40.
