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Line plots

Statistics and data · level 2, lesson 5 of 6

A question to start

A class of 20 children counted how many baby teeth each of them has lost so far. Here are the answers, in the order the children said them:

2, 3, 1, 4, 2, 0, 5, 3, 2, 8, 1, 4, 3, 2, 0, 5, 2, 3, 1, 4

What is the most common number of lost teeth? Did anyone lose a lot more than everybody else? Is there a number that nobody had?

It is hard to tell from a jumble of numbers. These answers are different from favourite pets or seasons: they are numbers that sit in order on a number line, 0, 1, 2, 3, and so on. For data like this there is a graph that keeps the numbers in their places. It is called a line plot, and you can draw one in a minute with nothing but X's.

The idea

Two kinds of data

So far the answers to our surveys have been choices, like dog, cat or fish. Choices have no order: dog is not "bigger" than cat. You can put the bars of a bar graph in any order.

But some answers are numbers that you count or measure: teeth lost, shoe size, how many letters in your name, how many minutes it takes to walk to school. These have a natural order. 3 teeth is more than 2 teeth and less than 4. A good graph for numbers keeps that order, with every number in its place, like the marks on a ruler.

Drawing a line plot

A line plot is a number line with an X above it for each answer.

  1. Draw a number line. Find the smallest answer (0) and the largest (8). Mark every number from the smallest to the largest, equally spaced: 0, 1, 2, 3, 4, 5, 6, 7, 8. Do not skip 6 and 7, even if nobody said them.
  2. Go through the answers one by one. For each answer, draw an X above its number, and cross the answer off the list. If there is already an X there, draw the new one just above it.
  3. Keep the X's the same size and stack them evenly, so the height of each stack shows how many.
  4. Add a title and a label under the line that says what the numbers are.

Here is the line plot for the lost teeth.

Baby teeth lost by 20 children 0 X X 1 X X X 2 X X X X X 3 X X X X 4 X X X 5 X X 6 7 8 X number of teeth lost

It is a little like a tally chart and a bar graph at once: each X is one child, and the stacks act like bars. But the number line keeps everything in order and shows the numbers nobody said.

What a line plot shows

Now the questions from the start are easy.

The most common value. The tallest stack is above 2, with 5 X's. So the most common number of lost teeth is 2. Grown-ups call the most common value the mode; you will meet it again in level 3.

Gaps. Above 6 and 7 there are no X's at all. A place with no X's is called a gap. Nobody in the class had lost 6 or 7 teeth.

An unusual value. One X sits all alone at 8, far from the rest, on the other side of the gap. A value that is far away from the others is called an outlier (it lies out away from the rest). One child has lost many more teeth than anyone else. Maybe they are older, or maybe they had an accident. Outliers are worth a second look: they might be a real surprise, or a mistake in the data.

Where most of the data sits. Most X's are piled up between 1 and 4. That is called a cluster: a bunch of values close together. Most children in this class have lost between 1 and 4 teeth.

How spread out. The answers go from 0 (smallest) to 8 (largest). That tells you how widely the answers are spread.

Why every number gets a mark

Imagine leaving out 6 and 7 because nobody said them, and putting 8 straight after 5. The plot would look as if the child with 8 teeth lost were close to everyone else. The gap, and the outlier, would disappear. A number line must be like a ruler: every number in its place, equally spaced. That is what lets the plot tell the truth about gaps and outliers.

Counting from a line plot

Each X is one child, so the number of X's is the number of children. Count them: 2 and 3 and 5 and 4 and 3 and 2 and 1 make 20. Twenty children, twenty X's. The total check works here too.

Worked examples

Example 1. Questions from the teeth plot.

  1. How many children have lost 3 teeth? Count the X's above 3: 4.
  2. How many have lost no teeth? Above 0: 2.
  3. How many have lost 4 or more? Above 4: 3. Above 5: 2. Above 8: 1. (Above 6 and 7: none.) 3 and 2 and 1 make 6 children.
  4. How many have lost fewer than 2? Above 0: 2, above 1: 3. That makes 5 children.
  5. Check: 6 children have 4 or more, 5 have fewer than 2, and 5 have exactly 2, and 4 have exactly 3. 6 and 5 and 5 and 4 make 20. Every child counted once.

Example 2. Shoe sizes. At a family party, 16 grown-ups told their shoe sizes. Here is the line plot.

Shoe sizes of 16 grown-ups 6 X X 7 X X X 8 X X X X X 9 X X X 10 X X 11 12 13 X shoe size
  1. Most common size: the tallest stack is above 8, with 5 X's.
  2. Gaps: no X's above 11 or 12.
  3. Outlier: one person wears size 13, far from the others.
  4. Cluster: most people wear sizes 6 to 10.
  5. How many wear size 9 or bigger? Above 9: 3, above 10: 2, above 13: 1. That makes 6 people.
  6. Why is this useful? A shoe store planning to stock shoes for these people would want lots of size 8, a few of the sizes on each side, and perhaps just one pair of size 13.

Example 3. Making a line plot. Twelve children counted the letters in their first names: 3, 5, 4, 4, 6, 5, 4, 7, 3, 4, 5, 10.

  1. Smallest 3, largest 10. Draw a number line with 3, 4, 5, 6, 7, 8, 9, 10, equally spaced.
  2. Put an X for each answer, crossing it off: 3 gets 2 X's, 4 gets 4, 5 gets 3, 6 gets 1, 7 gets 1, 10 gets 1.
  3. Check: 2 and 4 and 3 and 1 and 1 and 1 make 12.
Letters in our first names 3 X X 4 X X X X 5 X X X 6 X 7 X 8 9 10 X number of letters

Read it: the most common is 4 letters. Gaps at 8 and 9. The name with 10 letters is an outlier.

Try it

Use the teeth line plot for questions 1 to 3.

  1. How many children have lost exactly 5 teeth?
  2. How many children have lost 1 tooth or 2 teeth?
  3. A new child joins the class and has lost 6 teeth. What changes on the plot? Is there still a gap?
  4. Make a line plot of these numbers of pets per family: 1, 0, 2, 1, 1, 3, 0, 1, 2, 6. What is the most common number? Where is the gap? Is there an outlier?
  5. Leo made a line plot of minutes to walk to school, but his number line goes 5, 6, 7, 10, 15. What is wrong?
  6. Why is a line plot a good graph for shoe sizes, but not for favourite colours?
  7. In the shoe size plot, how many people wear a size smaller than 8?
Answers
  1. 2 children.
  2. Above 1: 3. Above 2: 5. 3 and 5 make 8 children.
  3. A new X appears above 6, so the 6 stack has 1 X. There is still a gap at 7. Now the child with 8 lost teeth is less far away from the rest.
  4. Number line 0, 1, 2, 3, 4, 5, 6. X's: 0 gets 2, 1 gets 4, 2 gets 2, 3 gets 1, 6 gets 1 (2 and 4 and 2 and 1 and 1 make 10). Most common: 1 pet. Gap: 4 and 5. Outlier: the family with 6 pets.
  5. The numbers are not equally spaced: from 7 to 10 is 3 minutes, and from 10 to 15 is 5 minutes, but they are drawn the same distance apart. He must mark every number from 5 to 15, equally spaced, like a ruler.
  6. Shoe sizes are numbers in order, so they fit on a number line, and gaps and outliers make sense. Colours have no order: there is no "between red and blue", so there is no number line to put them on. Use a bar graph for colours.
  7. Above 6: 2. Above 7: 3. 5 people.

Common mistake

Leaving out numbers that nobody chose. Every number from smallest to largest goes on the line, equally spaced, even if it has no X's. Otherwise the gaps vanish and outliers look closer than they are.

Drawing X's of different sizes, or stacking them unevenly. Then a tall stack might hold fewer X's than a short one. Keep every X the same size and the same distance apart.

Where it leads