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Analyze scatterplots, determine lines of best fit, interpret slope and intercepts in context, and make predictions using linear and nonlinear models.
Learn step-by-step with practice exercises built right in.
A scatterplot displays the relationship between two quantitative variables as points on a coordinate plane.
| Pattern | Description | Example |
|---|---|---|
| Positive | As increases, increases | Height vs. weight |
| Negative | As increases, decreases | Temperature vs. hot chocolate sales |
| None | No clear pattern | Shoe size vs. GPA |
The line of best fit is the straight line that best approximates the data.
"For each additional [unit of ], [the variable] is predicted to [increase/decrease] by [slope] [units of ]."
Example: If models the relationship between hours studied () and test score (): "For each additional hour of study, the test score is predicted to increase by 2.5 points."
"When [ variable] is 0, the predicted [ variable] is []."
Note: The -intercept may not always make practical sense (e.g., "0 hours of study" may be unrealistic).
A good model has residuals that are randomly scattered around zero. A pattern in residuals suggests the model is not a good fit.
| value | Meaning |
|---|---|
| Perfect positive linear relationship | |
| Perfect negative linear relationship | |
| No linear relationship | |
| $ | r |
| $ | r |
Remember:
"Which best describes the relationship?" → positive/negative, strong/weak, linear/nonlinear
"In context, what does the slope represent?" → rate of change per unit
Given a point and the line equation, calculate residual = actual − predicted.
Use the line equation to predict for a given value.
The point farthest from the line of best fit (largest residual).
A scatterplot shows the relationship between hours of sleep () and alertness score (). The points trend upward from left to right and cluster closely around a line. How would you describe this association?
Direction: Points trend upward → positive association
Strength: Points cluster closely → strong association
Form: Follows a line → linear association
Answer: Strong, positive, linear association.
In context: As hours of sleep increase, alertness scores tend to increase.
A scatterplot shows the relationship between hours of sleep () and alertness score (). The points trend upward from left to right and cluster closely around a line. How would you describe this association?
Direction: Points trend upward → positive association
Strength: Points cluster closely → strong association
Form: Follows a line → association
The line of best fit for a scatterplot relating years of experience () to salary in thousands () is . Interpret the slope in context.
The line of best fit for a scatterplot relating years of experience () to salary in thousands () is . Interpret the slope in context.
The line of best fit is . A data point has coordinates . What is the residual for this point?
The line of best fit is . A data point has coordinates . What is the residual for this point?
A researcher collects data and finds a correlation coefficient of between ice cream sales and drowning incidents. Can the researcher conclude that eating ice cream causes drowning?
Answer: No!
Explanation: A strong correlation () shows that ice cream sales and drowning incidents are — they tend to increase together. However, correlation does NOT prove causation.
A researcher collects data and finds a correlation coefficient of between ice cream sales and drowning incidents. Can the researcher conclude that eating ice cream causes drowning?
Answer: No!
Explanation: A strong correlation () shows that ice cream sales and drowning incidents are — they tend to increase together. However, correlation does NOT prove causation.
A line of best fit is for data where ranges from to . Which of the following predictions is most reliable?
A line of best fit is for data where ranges from to . Which of the following predictions is most reliable?
Answer: Strong, positive, linear association.
In context: As hours of sleep increase, alertness scores tend to increase.
The slope is 3.2.
Interpretation: For each additional year of experience, the predicted salary increases by $3,200 (3.2 thousand dollars).
Template: "For each 1-unit increase in [x-variable], the [y-variable] is predicted to [increase/decrease] by [slope] [units]."
Note: The -intercept of 32 means a person with 0 years of experience has a predicted salary of $32,000.
The slope is 3.2.
Interpretation: For each additional year of experience, the predicted salary increases by $3,200 (3.2 thousand dollars).
Template: "For each 1-unit increase in [x-variable], the [y-variable] is predicted to [increase/decrease] by [slope] [units]."
Note: The -intercept of 32 means a person with 0 years of experience has a predicted salary of $32,000.
Step 1: Find the predicted value at :
Step 2: Calculate the residual:
Answer: The residual is .
Interpretation: The actual value (88) is 3 units ABOVE the predicted value (85), so this point lies above the line of best fit.
Step 1: Find the predicted value at :
Step 2: Calculate the residual:
Answer: The residual is .
Interpretation: The actual value (88) is 3 units ABOVE the predicted value (85), so this point lies above the line of best fit.
What's really happening: There is a confounding variable — hot weather. When it's hot:
The heat is the common cause. Ice cream doesn't cause drowning.
SAT Rule: Only a randomized controlled experiment can establish causation. Observational studies can only show association.
What's really happening: There is a confounding variable — hot weather. When it's hot:
The heat is the common cause. Ice cream doesn't cause drowning.
SAT Rule: Only a randomized controlled experiment can establish causation. Observational studies can only show association.
A) Predicting when B) Predicting when C) Predicting when D) Predicting when
Key concept: Interpolation vs. Extrapolation
The data ranges from to .
A) : This is WITHIN the data range → interpolation → most reliable ✓ B) : Beyond the range → extrapolation → less reliable ✗ C) : Far beyond the range → extrapolation → unreliable ✗ D) : Below the range → extrapolation → unreliable ✗
Answer: A
SAT Tip: Predictions within the data range (interpolation) are more trustworthy than predictions outside it (extrapolation). The farther outside the range, the less reliable the prediction.
A) Predicting when B) Predicting when C) Predicting when D) Predicting when
Key concept: Interpolation vs. Extrapolation
The data ranges from to .
A) : This is WITHIN the data range → interpolation → most reliable ✓ B) : Beyond the range → extrapolation → less reliable ✗ C) : Far beyond the range → extrapolation → unreliable ✗ D) : Below the range → extrapolation → unreliable ✗
Answer: A
SAT Tip: Predictions within the data range (interpolation) are more trustworthy than predictions outside it (extrapolation). The farther outside the range, the less reliable the prediction.