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Scatter Plots
& Association

Learn to plot bivariate data, spot trends, and describe the direction, form, and strength of association between two variables.

Statistics

What Is Bivariate Data?

Bivariate data means you collect two measurements for each individual or item in your study. Each data point is an ordered pair \( (x, y) \) where \( x \) is the explanatory variable and \( y \) is the response variable.

\( x \): Explanatory
The variable you think might influence or predict the other. It goes on the horizontal axis.
\( y \): Response
The variable you think is affected or being predicted. It goes on the vertical axis.

What Is a Scatter Plot?

A scatter plot is a graph that displays bivariate data as a collection of dots on a coordinate grid. Each dot represents one ordered pair \( (x, y) \). The overall pattern of dots reveals the association between the two variables.

Hours Studied Test Score 1 2 3 4 5 60 70 80 90

Why Scatter Plots Matter

Scatter plots help you see relationships that a table of numbers hides. They answer a key question: "When one variable changes, does the other tend to change too?"

★ Reveal Trends
Spot patterns that let you predict one variable from another.
★ Identify Outliers
Find data points that don't fit the overall pattern.
★ Guide Decisions
Use the relationship to make informed choices in science, business, and everyday life.

Describing Association: Three Keys

Whenever you look at a scatter plot, describe the association using three characteristics:

1 Direction
Positive: both variables rise together. Negative: one rises while the other falls. None: no clear trend.
2 Form
Linear: dots cluster around a straight line. Non-linear: dots follow a curve.
3 Strength
Strong: dots are tightly packed around the trend. Weak: dots are widely scattered.

Direction: Positive, Negative, or None

Positive Negative No Association

Form: Linear vs Non-Linear

Linear Non-Linear

Strength: Strong vs Weak

Strong Weak

Correlation vs Causation

Correlation
Two variables move together in a pattern. The scatter plot shows an association (positive or negative).
Causation
Changing one variable directly causes the other to change. This requires a controlled experiment, not just observation.

Rule of thumb: Scatter plots show correlation. Never claim causation from a scatter plot alone.

Outliers & Clusters

Outlier Cluster A Cluster B
Outlier
A point that falls far away from the overall pattern. It may indicate an error or a genuinely unusual case.
Cluster
A group of points that bunch together, separate from other groups. Clusters may suggest subgroups in your data.

How to Create a Scatter Plot

1
Identify the variables. Decide which is explanatory \( (x) \) and which is response \( (y) \).
2
Set up axes. Label the horizontal axis with \( x \) and the vertical axis with \( y \). Choose a scale that fits all data values.
3
Plot each ordered pair. For every \( (x, y) \) in your table, place a dot at the matching coordinates.
4
Describe the association. State the direction, form, and strength. Note any outliers or clusters.

Example 1: Temperature & Lemonade Sales

Temperature (\(^{\circ}\text{F}\))
Cups Sold 60 70 80 90 10 20 30 40 50

Example 2: Age of Car & Value

Car Age (years)
Value (\(\$1000\text{s}\))
1 3 5 7 9 5 10 15 20 25

Example 3: Shoe Size & Test Score

Shoe Size Test Score 6 8 10 12 60 80 100

Common Mistakes

✕ Swapping Axes
Putting the response variable on the \( x \)-axis misrepresents the intended relationship, as convention dictates the explanatory variable is on the horizontal axis.
✕ Claiming Causation
Saying one variable "causes" a change in the other based only on a scatter plot, without a controlled experiment.
✕ Ignoring Outliers
Failing to mention or investigate unusual points that fall far from the pattern.

Association Summary Table

Feature What to Look For
Direction Positive / Negative / None Do dots rise, fall, or scatter randomly?
Form Linear / Non-linear Do dots follow a line or a curve?
Strength Strong / Moderate / Weak How tightly do dots cluster around the trend?
Outliers Present / Absent Any dots far from the pattern?
Clusters Present / Absent Any distinct subgroups of points?

Question 1 of 5

A scatter plot shows that as the number of hours of exercise per week increases, resting heart rate decreases. The dots are tightly packed. What is the best description of this association?

Question 2 of 5

Which variable should go on the \( x \)-axis of a scatter plot?

Question 3 of 5

A scatter plot of height vs shoe size for 50 adults shows dots rising from left to right but widely spread around the trend line. Which description best fits?

Question 4 of 5

A researcher finds a strong positive correlation between hours of TV watched and weight gain. Can they conclude that watching TV causes weight gain?

Question 5 of 5

Look at this scatter plot. What type of form does the association have?