Scatterplots and Line of Best Fit on the SAT

What Is a Scatterplot?

A scatterplot displays the relationship between two quantitative variables as points on a coordinate plane.

• $x$-axis: independent (explanatory) variable
• $y$-axis: dependent (response) variable

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Types of Association

| Pattern | Description | Example | |---|---|---| | Positive | As $x$ increases, $y$ increases | Height vs. weight | | Negative | As $x$ increases, $y$ decreases | Temperature vs. hot chocolate sales | | None | No clear pattern | Shoe size vs. GPA |

Strength of Association

• Strong: Points cluster tightly around a line/curve
• Weak: Points are widely scattered
• Nonlinear: Points follow a curve, not a line

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Line of Best Fit (Regression Line)

The line of best fit is the straight line that best approximates the data.

$y = mx + b$

• $m$ (slope): The predicted change in $y$ for each 1-unit increase in $x$
• $b$ ($y$-intercept): The predicted value of $y$ when $x = 0$

Interpreting Slope in Context

"For each additional [unit of $x$], [the $y$ variable] is predicted to [increase/decrease] by [slope] [units of $y$]."

Example: If $y = 2.5x + 10$ models the relationship between hours studied ($x$) and test score ($y$): "For each additional hour of study, the test score is predicted to increase by 2.5 points."

Interpreting the $y$-Intercept

"When [$x$ variable] is 0, the predicted [$y$ variable] is [$b$]."

Note: The $y$-intercept may not always make practical sense (e.g., "0 hours of study" may be unrealistic).

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Residuals

$\text{Residual} = \text{Actual value} - \text{Predicted value}$

• Positive residual: Actual > Predicted (point is ABOVE the line)
• Negative residual: Actual < Predicted (point is BELOW the line)
• Zero residual: Point is exactly ON the line

Residual Plots

A good model has residuals that are randomly scattered around zero. A pattern in residuals suggests the model is not a good fit.

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Correlation Coefficient ($r$)

| $r$ value | Meaning | |---|---| | $r = 1$ | Perfect positive linear relationship | | $r = -1$ | Perfect negative linear relationship | | $r = 0$ | No linear relationship | | $|r|$ close to 1 | Strong linear relationship | | $|r|$ close to 0 | Weak or no linear relationship |

Remember:

• $r$ only measures LINEAR relationships
• $r$ does NOT indicate causation
• Outliers can significantly affect $r$

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Making Predictions

Interpolation vs. Extrapolation

• Interpolation: Predicting within the range of data → generally reliable
• Extrapolation: Predicting beyond the data range → less reliable (may be inaccurate)

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SAT Question Types

Type 1: Describe the Association

"Which best describes the relationship?" → positive/negative, strong/weak, linear/nonlinear

Type 2: Interpret Slope or $y$-Intercept

"In context, what does the slope represent?" → rate of change per unit

Type 3: Find a Residual

Given a point and the line equation, calculate residual = actual − predicted.

Type 4: Make a Prediction

Use the line equation to predict $y$ for a given $x$ value.

Type 5: Identify an Outlier

The point farthest from the line of best fit (largest residual).

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Common SAT Mistakes

1. Claiming causation from a scatterplot — scatterplots show ASSOCIATION, not causation
2. Extrapolating too far beyond the data range
3. Misinterpreting slope — it's per unit change, not total change
4. Confusing positive and negative residuals — positive = above the line
5. Ignoring the context — always interpret slope and intercept in terms of the actual variables