Eighth Grade Mathematics: Linear, Nonlinear, Positive, and Negative Association
Once bivariate data is plotted on a scatter plot, the primary objective is to characterize the pattern of association between the two variables. In statistics, association describes the relationship or statistical dependency between two variables: knowing the value of one variable allows you to make informed predictions about the other. In this chapter, you will master classifying patterns of association across three distinct dimensions: direction (positive, negative, or none), form (linear or nonlinear), and strength (strong, moderate, or weak).
The Direction of Association: Positive versus Negative
The direction of association describes whether the two variables move together in the same direction or move in opposite directions:
1. Positive Association:
As the independent variable x increases, the dependent variable y tends to INCREASE.
The cloud of points rises from lower-left to upper-right (positive slope).
Examples: Height and shoe size; Study hours and exam scores; Engine size and fuel burn.
2. Negative Association:
As the independent variable x increases, the dependent variable y tends to DECREASE.
The cloud of points falls from upper-left to lower-right (negative slope).
Examples: Vehicle weight and gas mileage; Elevation and temperature; Practice time and golf score.
3. No Association:
There is no discernible relationship between x and y.
Changes in x do not predict changes in y; points scatter randomly with a flat trend.
Examples: Shoe size and IQ; Birthday month and running speed.
Visualizing Direction of Association:
Positive Association Negative Association No Association
y y y
| * * | * * | * * *
| * * | * * | * * * *
| * * | * * | * * *
| * * | * * | * * *
--+-------------> x --+-------------> x --+-------------> x
(Rises Left to Right) (Falls Left to Right) (Random Cloud)
The Form of Association: Linear versus Nonlinear
The form of association describes the overall geometric shape traced by the data points:
1. Linear Association:
The pattern of data points clusters tightly or loosely along a STRAIGHT LINE.
The rate of change between the variables is roughly constant.
2. Nonlinear Association:
The pattern of data points follows a CURVED PATH (such as a parabola, exponential curve,
or wave) rather than a straight line.
The rate of change varies as x increases.
Visualizing Form of Association:
Linear Form Nonlinear Form (Curved)
y y
| * | * *
| * * | \ /
| * * | * *
| * * | \___________/
--+-------------> x --+--------------------> x
Points follow a straight line Points follow a curved U-shape
A common example of nonlinear association is stopping distance versus driving speed: doubling your driving speed more than doubles your braking distance because kinetic energy scales with speed squared, creating a steep upward curve.
The Strength of Association: Strong versus Weak
The strength of an association describes how closely the data points cluster around the underlying trend line or curve:
1. Strong Association:
The data points cluster very tightly along the trend line with minimal scatter.
Predictions made from strong associations have high accuracy and confidence.
2. Weak Association:
The data points form a wide, dispersed cloud that shows a general upward or downward drift,
but with substantial variation away from the trend line.
Predictions made from weak associations carry high uncertainty.
Strong Positive Weak Positive
y y
| * | * *
| * * | * * * *
| * * | * * *
| * * | * * *
--+-------------> x --+-------------> x
(Tightly clustered) (Loosely dispersed cloud)
Chapter Exercises
Exercise 1: Classify the association shown in each of the following scenarios by its direction (positive, negative, or none) and describe the form: first, the speed of a runner versus the time it takes to finish a 5-kilometer race; second, the number of hours an electric heater runs versus total electricity cost; and third, the age of a student versus their telephone number.
Exercise 2: A scatter plot tracks the speed of a motorboat (x) and its fuel consumption per mile (y). At low speeds, fuel consumption is moderate; at medium speeds, fuel consumption drops to a minimum; and at high speeds, fuel consumption surges upward. Describe the form of this association, explaining whether a straight line of best fit would be appropriate.
Exercise 3: Explain in paragraph form the difference between a strong negative association and a weak negative association on a scatter plot, discussing point dispersion and prediction reliability.
Solutions and Explanations
Solution 1: In the first scenario, comparing running speed to race completion time: as a runner's speed increases, the time required to complete the 5-kilometer race decreases. Because the two variables move in opposite directions, this displays a Negative Association. The form is curved (nonlinear) because time equals distance divided by speed, which produces a rational curve. In the second scenario, comparing heater operating hours to electricity cost: as the heater runs for more hours, total electricity consumption and cost increase steadily. Because both variables increase together at a steady hourly rate, this displays a Strong Positive Linear Association. In the third scenario, comparing student age to telephone number: telephone numbers are randomly assigned numerical codes with zero dependence on a person's age. This displays No Association, producing a scattered cloud of points with no trend.
Solution 2: The association between the motorboat's speed and its fuel consumption per mile is Nonlinear (curved). The data points begin at a moderate height, slope downward to a trough at medium cruising speeds, and then curve steeply upward at high speeds, tracing a U-shaped parabolic curve. Because the direction of the relationship changes—decreasing at first and then increasing—the rate of change is not constant. A straight line of best fit would be completely inappropriate for this data; a straight line would fail to capture the curvature, drastically overestimating fuel consumption at medium speeds and underestimating it at the extremes.
Solution 3: On a scatter plot, both a strong negative association and a weak negative association display a general downward drift from upper-left to lower-right, indicating that higher values of the independent variable correspond to lower values of the dependent variable. The critical difference lies in the dispersion of the data points and the reliability of predictions. In a strong negative association, the plotted points cluster very tightly in a narrow band along the downward trend line, resembling a dense pencil line. Because there is very little scatter or variation away from the trend, you can predict the dependent variable with high accuracy and confidence. In contrast, in a weak negative association, the points are widely scattered across the plane in a broad, diffuse cloud. While you can detect a subtle downward tilt, the wide vertical spread of points means that any prediction made from the trend line carries substantial uncertainty and a large margin of error.