Lesson 2: Analyzing Patterns and Associations

Property

A linear association is a relationship between two variables where the data points on a scatter plot tend to follow a straight line. A nonlinear association exists when the data points follow a clear pattern, but it is a curve, not a straight line.

Examples

• Linear: The relationship between the number of hours worked and the amount of money earned. As hours increase, earnings increase at a constant rate, forming a straight-line pattern.
• Nonlinear: The relationship between the speed of a car and its fuel efficiency (miles per gallon). Fuel efficiency might increase with speed up to a certain point, then decrease, forming a curved pattern.
• Linear: The relationship between the side length of a square and its perimeter. The points form a perfect straight line since $P = 4s$.

Explanation

When analyzing data on a scatter plot, the first step is to observe the overall pattern. If the points seem to cluster around a straight line, the association is linear. If the points follow a distinct curve, the association is nonlinear. A trend line is only appropriate for modeling linear associations; a curve would be used for nonlinear ones.

Property

When analyzing scatter plots, it is important to describe the general relationship between two variables.

Positive Association: As $x$-values increase, $y$-values tend to increase (upward trend)

Property

The strength of an association in a scatter plot is determined by how closely the data points follow a discernible pattern.

• Strong Association: Data points are tightly clustered around a line or curve.
• Weak Association: Data points are loosely scattered around a line or curve.
• No Association: Data points show no discernible pattern.

Examples

• Strong Association: A scatter plot comparing the number of hours a student studies ($x$) and their exam scores ($y$). If the points form a tight band that rises from left to right, it indicates a strong positive association. This shows that students who dedicate more time to studying generally achieve higher exam scores, suggesting a clear and consistent relationship between study time and academic performance.

• Weak Association: A scatter plot comparing a person's age ($x$) and the number of books they read per year ($y$). If the points are widely spread but show a slight downward trend, it indicates a weak negative association. This suggests that while older individuals may tend to read slightly fewer books, the relationship is not strong and there is considerable variation among people.