Scatterplots and Association

11.1: Scatterplots and Association. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 1 of 63 parts Practice Flashcards

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Direction of association

Objective: Describe how two measured quantities tend to change together.

Prerequisite skills: coordinate graphs and qualitative trend descriptions. A positive association means larger x-values tend to accompany larger y-values. A negative association means larger x-values tend to accompany smaller y-values. No association means no clear systematic trend in the observed pattern. Common error: calling a decreasing pattern positive because its y-values are positive. Restrictions and interpretation: Association is a pattern, not proof of causation; strength and direction are separate features.

Formulas

direction is the sign of the trend

Worked examples

Try itA scatterplot includes points (1, 41), (2, 38), (3, 35), (4, 32), with small deviations around this pattern. What is the association?Show answer

As x increases by 1, the typical y-value decreases by about 3. That is a negative association. The signs of the y-values do not determine trend direction, and the plot alone does not prove cause. Therefore the requested answer is Negative.

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Direction of association

Objective: Describe how two measured quantities tend to change together.

Formulas

direction is the sign of the trend

Worked examples

Try itA scatterplot includes points (1, 41), (2, 38), (3, 35), (4, 32), with small deviations around this pattern. What is the association?Show answer

Residual interpretation

Objective: Measure an observation relative to a fitted prediction.

A residual is observed minus predicted output. A positive residual means the observed point is above the fitted graph; a negative residual means below. Residual size describes prediction error in output units. Common error: reversing predicted and observed. Restrictions and interpretation: Association is a pattern, not proof of causation; strength and direction are separate features.

Formulas

$residual=y_observed−y_predicted$

Worked examples

Try itA fitted model predicts 78 units, but the observed value is 83. What is the residual?Show answer

Residual=observed−predicted=83−78=5 units. The positive sign places the observation above the model. Therefore the requested answer is 5.

Association versus cause

Objective: Recognize an observational limitation.

A trend can arise because x influences y, y influences x, another variable influences both, or the sample is unrepresentative. Random assignment in a well-designed experiment is needed to support a causal interpretation. Common error: concluding cause from a strong scatterplot pattern alone. Restrictions and interpretation: Association is a pattern, not proof of causation; strength and direction are separate features. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.

Formulas

$association ≠ causation$

Worked examples

Try itIn an observational survey of 330 people, exercise time is associated with lower reported stress. Which conclusion is warranted from that design alone?Show answer

The survey observes existing behavior rather than randomly assigning exercise. Other factors or reverse direction may explain the pattern. The supported statement is association in the observed sample. Therefore the requested answer is The sample shows an association; causation is not established.

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