SAT Scatterplots & Models: Lines of Best Fit, Residuals & Prediction

Scatterplots help show how two quantitative variables relate.

SAT questions may ask about:

• direction;
• strength;
• lines of best fit;
• predictions;
• residuals;
• interpolation/extrapolation.

Positive association

As x increases, y tends to increase.

Negative association

As x increases, y tends to decrease.

No clear association

No obvious upward or downward pattern.

Correlation is not causation

A strong association does not prove one variable causes the other.

Example:

Ice cream sales and sunburn cases may both rise in summer.

Ice cream does not necessarily cause sunburn.

Line of best fit

A line of best fit summarizes the trend.

Suppose:

[ y=2.4x+10 ]

Slope:

[ 2.4 ]

Interpretation:

For each 1-unit increase in x, predicted y rises by 2.4 units.

Intercept:

[ 10 ]

Interpret only if x=0 makes sense in the context. Graphing a candidate line in Desmos can help you check whether the equation matches the plotted points.

Prediction

If x=20:

[ y=2.4(20)+10=58 ]

Predicted y = 58. This is the same substitution process used for any linear equation.

Residual

[ \text{residual}=\text{actual}-\text{predicted} ]

Predicted: 58

Actual: 63

Residual:

[ 63-58=5 ]

Positive residual means the actual value is above the model's prediction.

Interpolation

Prediction inside the observed x-range.

Usually safer.

Extrapolation

Prediction outside the observed range.

Less reliable.

Do not assume a linear pattern continues forever.

Original practice

A model predicts:

[ y=5x+12 ]

1

What does slope 5 mean?

2

Predict y when x=7.

3

If actual y at x=7 is 50, residual?

4

If observed x-values range from 2 to 10, is predicting at x=8 interpolation or extrapolation?

5

Is predicting at x=25 interpolation or extrapolation?

Answers

1. Predicted y rises by 5 for each 1-unit increase in x.
2. 47
3. 3
4. Interpolation
5. Extrapolation

Common traps

• saying correlation proves causation;
• reversing residual subtraction;
• giving the slope without units/context;
• trusting extrapolation too much.

Not every SAT model is a straight line — when the data curves, see our quadratic equations guide. For more Math skills, browse our SAT Math guides.

Bottom line

For scatterplots, always ask:

direction, model, prediction, or deviation from prediction?

That tells you what the question is really testing.

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