Lesson 3: Trend Lines and Linear Models

Property

A trend line (or line of fit) is a line drawn on a scatter plot that models the relationship between two variables. It shows the general direction of the data and is drawn to be as close to all the data points as possible.

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. This indicates that students who spend more time studying generally achieve higher scores, suggesting a clear relationship between study habits and academic performance.

• If the age of a car increases and its value tends to decrease, a trend line would have a negative slope, showing a negative association. This reflects that older cars typically lose value over time, demonstrating a predictable trend in the used car market.

Property

The strength of association describes how closely the data points in a scatter plot follow a trend line.

• Strong Association: Data points are tightly clustered around the trend line.
• Weak Association: Data points are widely scattered around the trend line.

Examples

• Strong Association: A scatter plot of daily study hours ($x$) versus test scores ($y$) shows points lying very close to a single upward-sloping line. This indicates a strong, predictable relationship where more study time consistently relates to higher scores.
• Weak Association: A scatter plot of a person''s age ($x$) versus the number of books they read per year ($y$) shows points that are widely spread out but may have a slight downward trend. This indicates a weak relationship, as age is not a strong predictor of reading habits.

Explanation

The strength of association tells you how reliable the trend line is as a model for the data. A strong association means the relationship between the two variables is consistent and predictable. A weak association indicates that while there may be a general trend, the relationship is less consistent and contains more variability. Assessing strength is a visual judgment about how tightly the points hug the line of fit.

Property

A linear trend line can be used to approximate the general direction of a data set that displays a nonlinear pattern. While the line will not perfectly fit the curved data, it provides a simple model of the overall increasing or decreasing trend.

Examples

• A scatter plot shows a car''s value ($y$) over time ($x$). The value drops quickly in the first few years and then more slowly, creating a curve. A trend line with a negative slope can be drawn to show the overall decreasing value, even though the relationship is not perfectly linear.
• A scatter plot shows a plant''s height ($y$) over several weeks ($x$). The growth is rapid at first and then levels off, forming a curve. A trend line with a positive slope can show the overall positive association between time and height.

Explanation

Sometimes, the relationship between two variables is not a perfect straight line; it might be a curve. Even in these cases, we can draw a trend line to show the general direction of the data. This line acts as a simple approximation, helping us see if the data is generally increasing or decreasing. It is important to note that while the trend line shows the overall pattern, it does not capture the nuances of the curved relationship.