Learn on PengienVision, Mathematics, Grade 8Chapter 4: Investigate Bivariate Data

Lesson 2: Analyze Linear Associations

In this Grade 8 enVision Mathematics lesson from Chapter 4, students learn how to analyze linear associations in bivariate data by creating scatter plots, drawing trend lines, and identifying whether associations are positive, negative, strong, weak, or nonlinear. Students practice distinguishing between linear and nonlinear associations and assess the strength of a relationship based on how closely data points cluster around a trend line. Real-world contexts like height versus arm span and ice cream sales versus temperature help students apply these concepts to interpret two-variable data sets.

Section 1

Drawing the Line of Fit

Property

A trend line (or line of best fit/regression line) is a straight line drawn on a scatter plot that models the relationship between two quantitative variables. It provides a one-dimensional summary of a bivariate (2-dimensional) data set by showing the general direction of the data.

To draw it, use the primary method of 'eye-balling' to find the line that minimizes the distance between each data point and that line. A line of best fit will have about the same number of points above and below it and may or may not pass through any of the data points.

Examples

  • If the number of hours studied increases and test scores also tend to increase, a trend line would have a positive slope, showing a positive association.
  • A scatter plot shows hours spent practicing piano versus number of mistakes made in a performance. The points trend downwards, so an 'eyeballed' line with a negative slope is drawn to show that more practice is associated with fewer mistakes.
  • Data is collected on daily temperature and the number of bottles of water sold at a park. The points on the scatter plot go up and to the right. An 'eyeballed' line with a positive slope summarizes this positive association.

Section 2

Form of Association: Linear vs. Nonlinear

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=4sP = 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.

Section 3

Describing Linear Associations

Property

A linear association is described by its direction (positive or negative) and its strength (strong or weak).

Examples

  • A scatter plot where points cluster tightly around a line rising from left to right shows a strong, positive linear association.
  • A scatter plot where points are loosely scattered around a line falling from left to right shows a weak, negative linear association.
  • A scatter plot where points are randomly scattered with no clear pattern shows no linear association.

Explanation

To fully describe the relationship between two variables in a scatter plot, you should comment on both direction and strength. The direction indicates whether the variables increase together (positive) or one decreases as the other increases (negative). The strength indicates how closely the data points follow a straight-line pattern. Combining these two descriptors provides a complete picture of the linear association.

Lesson overview

Expand to review the lesson summary and core properties.

Expand

Section 1

Drawing the Line of Fit

Property

A trend line (or line of best fit/regression line) is a straight line drawn on a scatter plot that models the relationship between two quantitative variables. It provides a one-dimensional summary of a bivariate (2-dimensional) data set by showing the general direction of the data.

To draw it, use the primary method of 'eye-balling' to find the line that minimizes the distance between each data point and that line. A line of best fit will have about the same number of points above and below it and may or may not pass through any of the data points.

Examples

  • If the number of hours studied increases and test scores also tend to increase, a trend line would have a positive slope, showing a positive association.
  • A scatter plot shows hours spent practicing piano versus number of mistakes made in a performance. The points trend downwards, so an 'eyeballed' line with a negative slope is drawn to show that more practice is associated with fewer mistakes.
  • Data is collected on daily temperature and the number of bottles of water sold at a park. The points on the scatter plot go up and to the right. An 'eyeballed' line with a positive slope summarizes this positive association.

Section 2

Form of Association: Linear vs. Nonlinear

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=4sP = 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.

Section 3

Describing Linear Associations

Property

A linear association is described by its direction (positive or negative) and its strength (strong or weak).

Examples

  • A scatter plot where points cluster tightly around a line rising from left to right shows a strong, positive linear association.
  • A scatter plot where points are loosely scattered around a line falling from left to right shows a weak, negative linear association.
  • A scatter plot where points are randomly scattered with no clear pattern shows no linear association.

Explanation

To fully describe the relationship between two variables in a scatter plot, you should comment on both direction and strength. The direction indicates whether the variables increase together (positive) or one decreases as the other increases (negative). The strength indicates how closely the data points follow a straight-line pattern. Combining these two descriptors provides a complete picture of the linear association.