Eighth Grade Mathematics: Constructing and Interpreting Scatter Plots
In previous grades, you studied univariate data—data involving a single variable, such as tracking student test scores on a dot plot or box plot. In the real world, however, scientific and social investigations usually explore how two different characteristics interact simultaneously: does study time correspond to higher exam grades, does outdoor temperature affect ice cream sales, or does a person's height relate to their shoe size? Data that involves two numerical measurements recorded from the same subject is called bivariate data. In this chapter, you will master constructing scatter plots on the Cartesian plane and interpreting bivariate relationships.
The Structure of Bivariate Data and Scatter Plots
Bivariate data is collected as paired numerical observations (x, y) where x represents the independent (explanatory) variable and y represents the dependent (response) variable.
The Scatter Plot:
A scatter plot is a graph of plotted points on a coordinate grid that displays the
relationship between two sets of paired bivariate quantitative measurements.
Each individual dot represents a single subject or observation.
Constructing a Scatter Plot: Study Hours vs. Exam Score
Student Study Hours (x) Exam Score (y) Coordinate Point (x, y)
A 1 65 (1, 65)
B 2 70 (2, 70)
C 3 75 (3, 75)
D 4 85 (4, 85)
E 5 92 (5, 92)
Exam Score (y)
100 | * (5, 92)
90 | * (4, 85)
80 | * (3, 75)
70 | * (2, 70)
60 | * (1, 65)
0 +-----+-----+-----+-----+-----+-----> Study Hours (x)
0 1 2 3 4 5
Unlike function graphs where points are connected by a solid line, the dots on a scatter plot remain separate and disconnected. Each point represents an actual empirical measurement.
Designing Scales and Labeling Axes
Constructing an effective scatter plot requires designing appropriate coordinate scales that display the data clearly without distortion:
Scale Construction Rules:
1. Identify Independent and Dependent Variables:
Place the explanatory variable on the horizontal x-axis and the response variable on the vertical y-axis.
2. Determine Ranges:
Find the minimum and maximum values for both x and y.
3. Choose Uniform Increments:
Space tick marks evenly across each axis (e.g., counting by 2s, 5s, 10s, or 100s).
The x-axis and y-axis do NOT need to have the same scale increments!
4. Use Scale Breaks Only When Necessary:
If data begins far from zero, a zigzag break symbol indicates compressed space.
5. Provide Descriptive Titles and Units:
Always label both axes with variable names and measurement units (e.g., "Temperature in Degrees Celsius").
Plotting points accurately on a well-scaled grid ensures that patterns and correlations become visually obvious.
Initial Visual Interpretation: Seeing the Forest, Not Just the Trees
When interpreting a scatter plot, avoid focusing narrowly on single data points. Instead, look at the overall cloud of points to identify overarching trends:
As you look from left to right along the horizontal axis, do the points generally drift upward, drift downward, or scatter randomly like stars in the night sky? If points drift upward, higher values of x tend to pair with higher values of y. If points drift downward, higher values of x tend to pair with lower values of y. If no pattern appears, the two variables operate independently.
Chapter Exercises
Exercise 1: The table provides data on outside temperature in degrees Fahrenheit (x) and the number of cups of hot cocoa sold at a ski lodge (y): at 10 deg, 85 cups; at 20 deg, 70 cups; at 30 deg, 55 cups; at 40 deg, 40 cups; and at 50 deg, 20 cups. Describe how to construct the coordinate grid, plot the five points, and describe the overall trend observed in paragraph form.
Exercise 2: A researcher measures the shoe sizes and reading speeds (words per minute) of thirty eighth-grade students and plots the results on a scatter plot. Describe what the resulting pattern of points would look like, explaining whether you would expect any association between the variables.
Exercise 3: Explain in paragraph form why the points on a scatter plot representing empirical scientific data are almost never arranged in a perfectly straight line, discussing natural variability and measurement error.
Solutions and Explanations
Solution 1: To construct the scatter plot, place the independent explanatory variable—Outside Temperature in degrees Fahrenheit—on the horizontal x-axis, using a scale from 0 to 60 with tick marks every 10 units. Place the dependent response variable—Hot Cocoa Cups Sold—on the vertical y-axis, using a scale from 0 to 100 with tick marks every 10 or 20 units. Plot the five coordinate pairs as individual dots: (10, 85), (20, 70), (30, 55), (40, 40), and (50, 20). Inspecting the plotted points reveals a clear downward trend from left to right: as the outside temperature increases, the number of hot cocoa cups sold consistently decreases. This displays a strong negative association, which aligns with physical common sense: customers buy less hot beverage when the weather is warm.
Solution 2: On a scatter plot displaying shoe size on the horizontal axis and reading speed in words per minute on the vertical axis for eighth-grade students, the points would form a cloud of randomly scattered dots with no discernible direction or pattern. There is no biological, cognitive, or educational connection between the physical size of an adolescent's feet and their neurological ability to decode and comprehend written text. A student with a small shoe size may be a fast or slow reader, and a student with a large shoe size may likewise read quickly or slowly. Because the cloud of points shows no upward or downward drift, the scatter plot indicates that there is no association (zero correlation) between shoe size and reading speed.
Solution 3: In pure mathematical equations like y = 2x + 1, every plotted point lands precisely on a single straight line because the equation defines a deterministic relationship with zero tolerance for deviation. However, empirical scientific data gathered from real-world observations involves natural variability and physical measurement limitations. When collecting data from human subjects, ecological systems, or physical experiments, dozens of secondary variables influence the outcome: individual metabolism, environmental fluctuations, differences in reaction time, and the precision limits of measuring instruments. Even when two variables share a powerful underlying linear relationship—such as height and weight in humans—people of the exact same height have different bone densities, muscle masses, and body compositions. Consequently, real-world data points scatter around the underlying trend line like a loose swarm of bees rather than forming a flawless line.