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Scatter Plots
on the SAT

Master how to read, interpret, and model data using scatter plots, lines of best fit, and correlation: key skills for the SAT Math section.

SAT Math

What Is a Scatter Plot?

A scatter plot is a graph that displays two quantitative variables as a collection of points on a coordinate plane. Each point represents one data observation with an \( x \)-value and a \( y \)-value.

Hours Studied Test Score 1 2 3 4 5 6 60 70 80 90 100

Key Vocabulary

Correlation
The strength and direction of the linear relationship between two variables.
Line of Best Fit
A straight line that best approximates the data, used for predictions.
Outlier
A data point that falls far from the overall pattern of the scatter plot.

Types of Correlation

Positive
As \( x \) increases, \( y \) increases. Points trend upward left to right.
Negative
As \( x \) increases, \( y \) decreases. Points trend downward left to right.
No Correlation
No clear pattern. Points are scattered randomly with no trend.
Positive Negative None

The Line of Best Fit

The line of best fit (also called the regression line or trend line) is the straight line that minimizes the sum of the squared vertical distances from all data points to the line. Its equation has the form:

x y run rise b

Interpreting Slope and Intercept

1
Slope \( m \): For each 1-unit increase in \( x \), the predicted \( y \) changes by \( m \) units. Include units and direction (increase or decrease).
2
Intercept \( b \): The predicted value of \( y \) when \( x = 0 \). Ask: does \( x = 0 \) make sense in context?

Correlation vs. Causation

Correlation
Two variables move together in a pattern. A scatter plot can show correlation.
Causation
One variable directly causes the other to change. Requires an experiment to prove.

How to Solve Scatter Plot Problems

1
Read the axes. Identify what \( x \) and \( y \) represent, including units.
2
Describe the trend. Is it positive, negative, or no correlation? Linear or curved?
3
Use the line of best fit. Plug in \( x \) to find \( \hat{y} \), or solve for \( x \) given \( \hat{y} \).
4
Interpret in context. State what the slope and intercept mean using the real-world units.
5
Identify outliers. Look for points far from the line: they may affect the model.

Example 1: Reading the Graph

Advertising ($1000s) Revenue ($1000s) 0 2 4 6 8 10 10 20 30 40 50 best fit

Example 2: Making a Prediction

Using the model \( \hat{y} = 4.4x + 7 \) from Example 1, predict the revenue when a company spends $12,000 on advertising (\( x = 12 \)).

1
Substitute: \( \hat{y} = 4.4(12) + 7 \)
2
Calculate: \( \hat{y} = 52.8 + 7 = 59.8 \)
3
Interpret: The predicted revenue is approximately $59,800.

Example 3: Residuals

A residual is the difference between the actual value and the predicted value:

+40 above line −40 below line

Strength of Correlation

The SAT may describe a correlation as strong, moderate, or weak. This refers to how tightly the points cluster around the line of best fit.

Strong Moderate Weak

Linear vs. Nonlinear Models

Linear
Data follows a straight-line pattern. Model: \( \hat{y} = mx + b \). Constant rate of change.
Nonlinear
Data curves (exponential, quadratic). A straight line is a poor fit. Rate of change varies.
Linear Nonlinear

Common Mistakes

✕ Confusing Correlation with Causation
A strong association does not mean one variable causes the other. Look for the word "causes" in wrong answer choices.
✕ Misreading the Axes
Always check axis labels and units. Swapping \( x \) and \( y \) leads to a backwards interpretation of slope.
✕ Extrapolating Too Far
Using the model far outside the data range can produce unreliable predictions. Note the domain of the data.

Scatter Plot Summary

Concept What to Do
Trend Positive, negative, or none Look at overall direction of points
Strength Strong, moderate, or weak How tightly points cluster around the line
Slope \( m \) in \( \hat{y} = mx + b \) Interpret as rate of change with units
Intercept \( b \) in \( \hat{y} = mx + b \) Predicted \( y \) when \( x = 0 \)
Residual \( y - \hat{y} \) Actual minus predicted: above or below line
Prediction Plug in \( x \) Stay within the data range when possible

Question 1 of 5

A scatter plot shows that as the number of hours of exercise per week increases, resting heart rate decreases. Which best describes the association?

Question 2 of 5

The line of best fit for a scatter plot is \( \hat{y} = 2.3x + 15 \), where \( x \) is years of experience and \( y \) is hourly wage in dollars. What does the slope mean in this context?

Question 3 of 5

Using the model \( \hat{y} = 2.3x + 15 \), what is the predicted hourly wage for a worker with 10 years of experience?

Question 4 of 5

A data point has an actual value of \( y = 50 \) and the line of best fit predicts \( \hat{y} = 43 \). What is the residual?

Question 5 of 5

A study finds a strong positive correlation between ice cream sales and sunburn rates. Which conclusion is most appropriate?