Lesson 6: Analyzing Lines of Fit

Property

A line of best fit, also known as a trend line, is a straight line that best represents the data on a scatter plot.
This line may pass through some of the points, none of the points, or all of the points.
It is typically expressed in the form $y = mx + b$.

Examples

• Given a scatter plot of study hours vs. test scores, the line of best fit shows the general trend, such as scores increasing as study hours increase.
• For data on a car''s age and its value, the line of best fit would likely have a negative slope, showing the value decreases as the car gets older.
• A line of best fit can be visually estimated or calculated precisely using a method called linear regression.

Explanation

The line of best fit is a tool used to model the relationship between two variables in a data set. It provides a simple, linear approximation of the data''s overall pattern or trend. The line is positioned to be as close as possible to all the data points collectively, minimizing the overall distance from the points to the line.

Property

Linear regression is a statistical method for finding the line of best fit for a set of data points.
The method calculates the values for $a$ and $b$ in the equation $y = ax + b$ that minimize the sum of the squared residuals.

Examples

• A real estate agent uses linear regression to model the relationship between a house''s square footage and its selling price.
• A biologist might use linear regression to analyze the connection between daily temperature and the growth rate of a certain plant.

Explanation

Linear regression is the mathematical process used to determine the "line of best fit" for a scatter plot. This line is the one that comes closest to all of the data points simultaneously. By finding this line, we can model the linear relationship between two variables and make predictions. The method works by minimizing the overall error, specifically the sum of the squared vertical distances from each point to the line.

Property

A residual is the difference between an actual data point and the predicted value from the line of best fit:

A residual plot graphs the residuals against the $x$-values to evaluate how well the linear model fits the data.

Property

Using a regression line to estimate values between known data points is called interpolation.

Examples

Cocoa sales data is collected for temperatures between $2^\circ$C and $18^\circ$C. Using the regression line $C = -2.5T + 52$ to predict sales at $9^\circ$C is interpolation. We get $C = -2.5(9) + 52 = 29.5$ cups.

A baby whale's length is recorded at birth (0 months) and 7 months. Using a linear model to estimate its length at 4 months is interpolation, because 4 is between 0 and 7.