8th Grade · 8.SP.A.1

Scatter plots in 8th grade

In 8th grade math, scatter plots show how two measurements of the same people or things relate, such as hours studied and quiz scores. Your child learns to make a scatter plot from a table and to describe the pattern the dots make. Words like association, cluster and outlier become part of math class.

What the standard says

The official Common Core wording, quoted exactly. The plain version follows.

8.SP.A.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

What it means, in plain words

A scatter plot is a graph made of dots. Each dot stands for one person or thing that was measured in two ways. For a group of students, the x-axis might show hours studied and the y-axis the quiz score. A student who studied 3 hours and scored 7 becomes the dot at $(3, 7)$ . The dots are not joined with lines.

Lessons on scatter plots and association then ask what the pattern means. Association is how one measurement tends to change when the other one changes. In a positive association, both go up together, like height and shoe size. In a negative association, one goes up while the other goes down, like a car's age and its price. If the dots are spread out with no direction, there is no association.

Your child also describes the shape of the dots. In a linear association, the dots lie close to a straight line. In a nonlinear association, they follow a curve. A cluster is a group of dots bunched close together. An outlier is a dot far away from the rest, such as one student who studied a lot and still scored low.

Your child should be able to:

Make a scatter plot from a table, with one measurement on each axis
Describe an association as positive, negative or none
Tell a linear association from a nonlinear one
Point out clusters and outliers and say what they mean in the situation

Example

Five students recorded hours of piano practice in one week and mistakes at their recital: 1 hour and 9 mistakes, 2 and 7, 3 and 6, 4 and 4, 5 and 2. Make a scatter plot and describe the association.

Put hours of practice on the x-axis and mistakes on the y-axis.
Plot one dot per student: $(1, 9)$ , $(2, 7)$ , $(3, 6)$ , $(4, 4)$ and $(5, 2)$ .
From left to right the dots go down, so more practice goes with fewer mistakes.
The dots lie close to a straight line, so the scatter plot shows a negative linear association.

Example

Six students' shoe sizes and reading scores are $(5, 70)$ , $(6, 85)$ , $(7, 60)$ , $(8, 90)$ , $(9, 65)$ and $(10, 80)$ . What association does a scatter plot of this data show?

Plot shoe size on the x-axis and reading score on the y-axis.
As shoe size goes up, the scores jump up and down with no direction.
There is no association. Knowing a student's shoe size does not help you guess the reading score.

Where children usually get stuck

Your child joins the dots from left to right with a line, like a line graph.
Most graphs your child has made before connect their points, so joining them feels like finishing the graph. In a scatter plot each dot is a separate person, and nothing happens between two dots.
What helps. Point to the space halfway between two dots and ask who that would be. No student is there, so there is nothing to connect.
Your child puts student numbers 1, 2, 3 and so on along the x-axis and plots only the quiz scores.
The table lists the students in order, so the order looks like a measurement. A scatter plot needs two measurements for each student.
What helps. Ask which two things were measured for each person in the table. Those two go on the axes, and the student number goes nowhere.
Your child says a plot where the dots go down shows no relationship.
The child links association with dots that go up. A downward pattern is a real association, a negative one.
What helps. Look at the piano example together and ask: if you know the hours of practice, can you guess the mistakes? If the answer is yes, there is an association, in either direction.

Try four practice questions

Real 8th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

1Warm-up
Select all scatter plots that show a nonlinear association.
A
Plot A
B
Plot B
C
Plot C
D
Plot D
Show the answer and how to solve it
The only nonlinear plot is Plot A because the rate of increase accelerates from start to finish, forming a curve instead of a straight line.
1. 1Check whether the pattern bends or stays straight. A nonlinear association curves or changes steepness as it moves across the plot; a linear association keeps a roughly constant rate of change.
2. 2Plot A is nonlinear. The points start rising very slowly near (1, 1.0) and then sweep upward much more steeply toward (6, 7.0), so the pattern clearly curves rather than staying in a straight line.
3. 3Plots B, C, and D should not be selected. Each moves at a roughly steady pace: B rises evenly, C falls evenly, and D rises evenly. All three follow straight-line trends.
Common mistake. A common mistake is selecting B or D because they also go up. Going up does not make a plot nonlinear; only bending or changing steepness does. Plot A is the only one whose spacing between consecutive points grows larger and larger.
The idea underneath. Linear vs nonlinear depends on whether the rate of change stays about constant, so the points look straight instead of bending.
2On level
Look at the scatter plot. Which region best describes where a cluster of points appears?
A
Near x = 1 to 3 and y = 1 to 3
B
Near x = 6 to 8 and y = 7 to 9
C
Near x = 7 to 9 and y = 1 to 3
D
Near x = 4 to 6 and y = 4 to 6
Show the answer and how to solve it
The answer is Near x = 1 to 3 and y = 1 to 3 because that window contains the only clear 5-point cluster.
1. 1Look for the region where the points are most packed together, not just a place with one or two points.
2. 2The tightest group is the 5 points near (1,1), (1,2), (2,1), (2,2), and (2,3). That region is near x = 1 to 3 and y = 1 to 3.
3. 3The other regions are not clusters: near x = 6 to 8 and y = 7 to 9 there are only 2 points, near x = 7 to 9 and y = 1 to 3 there is only 1 point, and near x = 4 to 6 and y = 4 to 6 there is only 1 point.
4. 4So the best description is Near x = 1 to 3 and y = 1 to 3.
Common mistake. A common mistake is choosing the pair near (6,8) and (7,8). A cluster should be the clearest concentration of several nearby points, not just any 2 points that happen to be close.
The idea underneath. A cluster is identified by relative density: it is the region where points are more concentrated than the surrounding parts of the scatter plot.
3On level
A school researcher is making a scatter plot for a group of students to compare hours of studying per week and test score. Choose which variable belongs on each axis.
x-axis
y-axis
Show the answer and how to solve it
x-axis: hours of studying per week. y-axis: test score. These axes show the comparison; they do not prove that one variable causes the other.
1. 1Look at the two variables: hours of studying per week and test score.
2. 2For this comparison, hours of studying per week is a natural choice for the explanatory variable, the quantity used to look for a pattern in test score.
3. 3Put the explanatory variable on the x-axis. Here, that is hours of studying per week.
4. 4Put the other variable on the y-axis. Here, that is test score.
Common mistake. Do not choose an axis just because a measurement seems more important. Here, use hours of studying per week as the explanatory variable and choose each variable once.
The idea underneath. The explanatory variable is the quantity used to look for a pattern in another quantity. It usually goes on the x-axis. The other quantity goes on the y-axis.
4Stretch
Look at the table. The first column is a student ID number, not a measurement. What type of graph would best show the relationship between the two measurements for each student?
Student ID Height (cm) Arm Span (cm)
101 138 140
102 143 142
103 149 151
104 155 153
105 161 163
106 166 165
107 172 174
A
scatter plot
B
bar graph
C
dot plot
D
line graph
Show the answer and how to solve it
The answer is scatter plot because each student has two numerical measurements, Height (cm) and Arm Span (cm), that can be plotted together as one point.
1. 1Look at the table. Each row is one student. The first column gives that student's ID number — it is just a label, not a measurement on any axis.
2. 2The two measurement columns are Height (cm) and Arm Span (cm). For example, student 101 has height 138 and arm span 140. Those two numbers are recorded together for the same person, so that row could become the point (138, 140) on a graph.
3. 3A scatter plot is designed for paired numerical data — each pair becomes one point. A bar graph compares category totals, a dot plot shows one list of numbers, and a line graph fits data ordered over time. None of those match two paired measurements per case.
4. 4The IDs 101–107 may look numerical, but they do not belong on either axis. Only Height (cm) and Arm Span (cm) are the true measurements, so a scatter plot is the correct choice.
Common mistake. A common mistake is choosing line graph because the student ID numbers 101 through 107 look like a numerical sequence. Those IDs are only labels; the paired measurements are Height (cm) and Arm Span (cm).
The idea underneath. When each row gives two numerical measurements for the same case, that row becomes one point on a scatter plot.

Student ID	Height (cm)	Arm Span (cm)
101	138	140
102	143	142
103	149	151
104	155	153
105	161	163
106	166	165
107	172	174

Student ID	Height (cm)	Arm Span (cm)
101	138	140
102	143	142
103	149	151
104	155	153
105	161	163
106	166	165
107	172	174

Student ID	Height (cm)	Arm Span (cm)
101	138	140
102	143	142
103	149	151
104	155	153
105	161	163
106	166	165
107	172	174