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Scatter Plots
Patterns in Data

Learn to construct scatter plots, identify associations, and describe patterns like clustering, outliers, and linear vs. nonlinear trends.

8th Grade Math

What Is a Scatter Plot?

A scatter plot is a graph that displays two numerical variables as a collection of points on a coordinate plane. Each point represents one observation, with the \(x\)-coordinate for one variable and the \(y\)-coordinate for the other.

Bivariate Measurement Data

Bivariate data means we are measuring two variables for each individual or observation. We call them the explanatory variable (\(x\)) and the response variable (\(y\)).

Explanatory (\(x\))
The variable we think might influence the other. Plotted on the horizontal axis.
Response (\(y\))
The variable we think might be affected. Plotted on the vertical axis.

Why Scatter Plots Matter

Scatter plots let us see relationships that tables of numbers can hide. They help us spot trends, clusters, and outliers at a glance.

Reveal Trends
See if values go up together, down together, or show no pattern at all.
Spot Outliers
Identify unusual data points that don't fit the overall pattern.
Make Predictions
Use the trend to estimate values we haven't measured yet.

Positive vs. Negative Association

Positive Association \(x\) \(y\) Negative Association \(x\) \(y\)
Positive
As \(x\) increases, \(y\) also increases. Points rise from left to right.
Negative
As \(x\) increases, \(y\) decreases. Points fall from left to right.

No Association in Data

\(x\) \(y\) No Association

Linear vs. Nonlinear Association

Linear Nonlinear
Linear
Points cluster around a straight line. The rate of change is roughly constant.
Nonlinear
Points follow a curved pattern. A straight line would miss the trend.

Clustering and Outliers

\(x\) \(y\) Cluster A Cluster B Outlier
Clustering
Groups of points that bunch together. May reveal subgroups in the data.
Outliers
Points far from the overall pattern. They may indicate errors or special cases.

Correlation vs. Causation

A scatter plot can show that two variables are associated (correlated), but that does not prove one causes the other.

Correlation
Two variables move together: as one changes, the other tends to change in a predictable way.
Causation
One variable directly causes the change in the other. Requires more evidence than a scatter plot alone.

How to Construct a Scatter Plot

1
Identify the two variables. Decide which is \(x\) (explanatory) and which is \(y\) (response).
2
Set up the axes. Label each axis with the variable name and choose an appropriate scale.
3
Plot each data pair. For each \((x, y)\) pair, place a dot at the corresponding coordinates.
4
Title your graph. Add a descriptive title that tells the reader what the scatter plot shows.
5
Describe the pattern. State the direction, form, strength, and any outliers or clusters.

Example 1: Study Hours vs. Score

0 1 2 3 4 5 6 Study Hours 50 60 70 80 90 100 Test Score

Example 2: TV Hours vs. Grade

0 1 2 3 4 Hours of TV per Day 60 70 80 90 Grade (%)

Example 3: Growth with an Outlier

Weeks of Practice Typing Speed (WPM) 0 2 4 6 8 10 12 10 25 40 55 70 Outlier

Describing Patterns Checklist

1
Direction: Is the association positive, negative, or none?
2
Form: Is the pattern linear (straight) or nonlinear (curved)?
3
Strength: Are points tightly clustered around the trend (strong) or widely scattered (weak)?
4
Outliers: Are there any points that fall far from the overall pattern?
5
Clusters: Do the data form distinct groups or subsets?

Common Mistakes

✕ Swapping Axes
Placing the response variable on the \(x\)-axis instead of the \(y\)-axis reverses the interpretation.
✕ Assuming Causation
Seeing a trend and claiming one variable causes the other. Correlation does not imply causation.
✕ Ignoring Outliers
Skipping over outliers without mentioning them. Always acknowledge and try to explain unusual points.

Summary of Associations

Pattern What It Looks Like Real-World Example
Positive Linear Points rise left to right near a line Height vs. shoe size
Negative Linear Points fall left to right near a line Price vs. demand for a product
Nonlinear Points follow a curve (not a line) Age vs. height (infancy to adulthood)
No Association Points scattered with no pattern Shoe size vs. test score
Clustering Points form distinct groups Income vs. spending (urban vs. rural)

Question 1 of 5

A scatter plot shows points that rise from left to right and cluster tightly around a straight line. How would you best describe this association?

Question 2 of 5

In a scatter plot comparing hours of exercise per week and resting heart rate, you expect the association to be:

Question 3 of 5

A data point in a scatter plot is far away from all other points and does not follow the trend. This point is called:

Question 4 of 5

A scatter plot of car age (\(x\)) vs. resale value (\(y\)) shows points that decrease steeply at first and then level off. This pattern is best described as:

Question 5 of 5

A scatter plot shows a strong positive association between number of firefighters sent to a fire and the amount of damage. Can we conclude that sending more firefighters causes more damage?