Create and Interpret Dot Plots | Grade 6 Math | LevelKite

You know that a statistical question has answers that vary. Those answers form a distribution, which is the way the data values are spread out. A dot plot is one way to draw the distribution of numerical data. Some books call it a line plot.

A dot plot is a number line with marks above it. The numbers along the line are the scale, and each number on the scale is a position. The scale goes up by the same amount each time. That amount is the step size.

Each mark stands for one observation, which is one recorded data value. A mark sits above the position that equals its value. The marks above one position form a stack. People often draw the marks as dots, and that gives the plot its name.

Here is a small set of data. Eight students each said how many pets they have:

$0, 0, 1, 1, 1, 2, 4, 4$

The unit is pets, and each mark is one student's answer. Look at the dot plot below. Its scale goes from $0$ to $4$ pets in steps of $1$ .

The two answers of $0$ are the two marks above $0$ .
The three answers of $1$ are the three marks above $1$ .
The one answer of $2$ is the one mark above $2$ .
Nobody answered $3$ , so the position for $3$ stays on the line with no marks. That is an empty stack.
The two answers of $4$ are the two marks above $4$ .

Now count the marks: $(2 + 3) + (1 + 0 + 2) = 5 + 3 = 8$ . That matches the $8$ students.

Line plot

Each mark is one student. From

0

to

4

pets, the stacks hold

2

,

3

,

1

,

0

and

2

marks.

Key concept

One mark for each observation

A dot plot shows every observation by itself. Each mark is one observation, and it sits above its exact value on the number line. Each value has its own position, so the plot never groups nearby values together.

The number of marks in a stack is the count of observations with that value. In this lesson, the scale starts at $0$ and goes up in equal steps. The plot shows every position on the scale, even when its stack is empty.

Because each mark is one observation, the number of observations is the sum of the counts of all the stacks. Counting the positions that have marks answers a different question: how many different values appear.

\text{number of observations} = \text{sum of the counts of all the stacks}

To create a dot plot, you turn a list of data into stacks of marks. In this lesson, you build every plot the same way.

Find the greatest value in the data. The scale starts at $0$ and must reach that value.
Choose a step size that lets every data value land exactly on a position. Whole-number data can use steps of $1$ . Data that are all multiples of $5$ can use steps of $5$ .
Draw the number line. Show every position from $0$ to the end of the scale, and label the line with its unit.
Count how many times each value appears. Put that many marks above the value. When the count is $0$ , leave the stack empty but keep the position.
Check your plot. The counts of all the stacks must add up to the number of data values, because each mark is one observation.

Why keep an empty position? A number line needs equal spacing. If you drop a value that nobody had, the values on each side look closer than they really are. The empty stack also tells a true fact: no observation has that value.

Worked example

Example 1: Create a dot plot from a list

Twelve students each recorded how many books they read over the summer: $3, 5, 2, 4, 3, 7, 2, 3, 5, 3, 2, 4$ . Create a dot plot of the data. The unit is books, and each mark stands for one student.

Find the greatest value, because the scale must reach it. The greatest value is 7 books. Every value is a whole number, so steps of 1 land exactly on every value.
Draw the number line from 0 to 7 in steps of 1, and label it books. With the position for 0, that makes 8 positions. The scale starts at 0, so 0 and 1 are on the line even though no student gave those answers.
$0, 1, 2, 3, 4, 5, 6, 7$
Count how many times each value appears, because each stack must show that count. Cross off each value as you count it, so you use every value exactly once. The value 2 appears three times, 3 appears four times, 4 appears twice, 5 appears twice and 7 appears once. The values 0, 1 and 6 do not appear, so their counts are 0.
$\text{counts from } 0 \text{ to } 7 \text{ books: } 0, 0, 3, 4, 2, 2, 0, 1$
Draw one mark for each observation, above its exact value. Stack three marks above 2, four marks above 3, two marks above 4, two marks above 5 and one mark above 7. Leave the stacks above 0, 1 and 6 empty, but keep those positions on the line.
Check the plot. Each mark is one observation, so the counts of all 8 stacks must add up to the 12 values in the list.
$(0 + 0 + 3 + 4) + (2 + 2 + 0 + 1) = 7 + 5 = 12$

Answer

The dot plot has a scale from $0$ to $7$ books in steps of $1$ . From $0$ to $7$ books, its stacks hold $0$ , $0$ , $3$ , $4$ , $2$ , $2$ , $0$ and $1$ marks. That is $12$ marks for the $12$ students.

Line plot

The finished plot from Example 1. The stacks above

0

,

1

and

6

books are empty, and the

12

marks match the

12

students.

To interpret a dot plot, you read the stacks back into facts about the data. It is creating in reverse.

Value: Look at the position under a stack. When the scale starts at $0$ and has equal steps, the value of a position is its number of steps from $0$ times the step size.
Count: The number of marks in a stack is the number of observations with that value. An empty stack means no observation has that value.
Total: Each mark is one observation, so the number of observations is the sum of the counts of all the stacks.
Groups: To count the observations that fit a description, such as at least $4$ , add the counts of every stack that fits. At least $4$ means $4$ or more.

Try it on the books plot above. The stack above $3$ is taller than every other stack, so $3$ books is the most common answer. That stack has $4$ marks, so $4$ students read $3$ books. At least $4$ books means the stacks above $4$ , $5$ , $6$ and $7$ . Their counts add to $2 + 2 + 0 + 1 = 5$ , so $5$ students read at least $4$ books.

Worked example

Example 2: Read values, counts and the total

A quiz has $6$ questions worth $5$ points each, with no partial credit. Every score is a multiple of $5$ , from $0$ up to $6 \times 5 = 30$ points. The dot plot below shows the scores of one class. Each mark is one student. What is the lowest score? How many students scored exactly $20$ points? How many students took the quiz?

Read the scale first, because it tells you the value under each stack. It starts at 0 and goes up in steps of 5 points, so the positions are 0, 5, 10, 15, 20, 25 and 30. From 0 to 30 points, the stacks hold 0, 1, 0, 3, 6, 5 and 3 marks.
Find the lowest score. Starting from 0, it is under the first stack that has marks, because an empty stack means no student has that score. The stack above 0 is empty, and the stack above 5 has 1 mark.
$\text{lowest score} = 5 \text{ points}$
Find the position for 20 points. The scale starts at 0 and has equal steps of 5, so the position 4 steps from 0 has the value 4 times 5, which is 20. The stack above it has 6 marks, and each mark is one student.
$4 \times 5 = 20$
Find the total. Each mark is one observation, so add the counts of all 7 stacks. An empty stack adds 0.
$(0 + 1 + 0 + 3) + (6 + 5 + 3) = 4 + 14 = 18$

Answer

The lowest score is $5$ points. Six students scored exactly $20$ points, and $18$ students took the quiz.

Line plot

Quiz scores in steps of

5

points. From

0

to

30

points, the stacks hold

0

,

1

,

0

,

3

,

6

,

5

and

3

marks, which is

18

marks in all.

Worked example

Example 3: Combine stacks to answer questions

Use the quiz dot plot above. Each mark is one student, and the scale goes from $0$ to $30$ points in steps of $5$ . How many students scored at least $20$ points? How many more students scored $25$ points than scored $15$ points?

At least 20 points means 20 points or more. On this scale, those scores are 20, 25 and 30. Add the counts of those three stacks, because each mark in them is one student who fits.
$6 + 5 + 3 = 11 + 3 = 14$
Check with the rest of the plot. Every student scored either below 20 or at least 20, so the two groups must add up to the total of 18 from Example 2. The stacks for scores below 20 are above 0, 5, 10 and 15.
$(0 + 1 + 0 + 3) + 14 = 4 + 14 = 18$
Compare two stacks. The stack above 25 has 5 marks, and the stack above 15 has 3 marks. Subtract the counts to find how many more.
$5 - 3 = 2$

Answer

Of the $18$ students, $14$ scored at least $20$ points. Two more students scored $25$ points than scored $15$ points.

Takeaway. A dot plot keeps every observation in view: each mark is one observation above its exact value. In this lesson, you start the scale at $0$ , show every position in equal steps, and leave a stack empty when no observation has that value. To interpret a plot, read each value from its position and each count from the number of marks in the stack. The number of observations is the sum of the counts of all the stacks, while the number of positions that have marks is only the number of different values.

Next, you will create and interpret histograms, a display that groups data values instead of showing each one by itself.