Lesson 11-1: Scatter Plots

Property

Bivariate data consists of pairs of values for two different variables. A scatter plot is a graph that displays these pairs of data as points $(x, y)$ on a coordinate plane to show the relationship between the two variables.

• $x$-axis: Represents the independent variable (the input or explanatory variable).
• $y$-axis: Represents the dependent variable (the output or response variable).
• Axis Break: A symbol used on an axis to indicate that a range of values starting from 0 has been skipped to avoid empty space.

Examples

• For data comparing "Hours Studied" and "Test Score", "Hours Studied" is the independent variable ($x$-axis) and "Test Score" is the dependent variable ($y$-axis). A single data point could be $(3, 85)$, representing 3 hours of study and a score of 85.
• If a dataset of weights ranges from 120 lbs to 160 lbs, place an axis break between 0 and 120 on the axis, then use consistent increments of 10.

Explanation

To construct a scatter plot, first determine which variable is independent ($x$-axis) and which is dependent ($y$-axis). Next, choose a scale with consistent increments that covers the entire range of your data. If data values start far from zero, an axis break skips the empty space, making the pattern in the data much easier to see. Finally, plot each observation as an ordered pair $(x, y)$.

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

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.