Lesson 6.9

Line of Best Fit

Scatter plots are just dots. To make predictions, we need to turn those dots into a line. This line is our "best guess" or model of the real world.

Introduction

A "Line of Best Fit" (or Trend Line) is a straight line that best represents the data on a scatter plot. It may pass through some of the points, none of the points, or all of the points. Its job is to average out the noise.

Past Knowledge

Lesson 6.8 (Scatter Plots). You need to be able to identify the general direction (correlation) first.

Today's Goal

Draw a line through the middle of the "cloud" and write its equation.

Future Success

This is "Linear Regression". Your calculator can do it perfectly, but you need to understand the logic.

Key Concepts

How to Draw It

Visualize the Oval: Imagine drawing an oval around all your data points.
Slice the Oval: Draw a straight line through the long axis of the oval.
Balance: Try to have roughly the same number of points above the line as below it.
Pick Points ON THE LINE: To write the equation, do NOT use the original data points (unless the line happens to hit them). Pick two points exactly on your new line.

Worked Examples

Example 1: Estimating the Equation

Basic

Find the equation of the trend line.

Step 1: Pick Points on Line

The line passes nicely through and .

Note: Neither of these are original data points!

Step 2: Find Slope

Step 3: Write Equation

In , students often guess the slope is 2. It isn't! You always have to solve for first ().

Example 2: Making Predictions

Application

Use the equation from Example 1 to predict the value when .

Step 1: Plug it in

This is "Extrapolation"—predicting outside your data range.

Example 3: Interpreting the Model

Analysis

In the equation , what do the numbers actually mean?

Slope ()

This is the Rate of Change.

"For every 1 unit increase in x, y increases by 0.8."

Y-Intercept ()

This is the Starting Value.

"When x is 0, y is 1."

Common Pitfalls

Connect the Dots

Do NOT just draw a line from the very first dot to the very last dot. That usually ignores the middle of the trend. The line should go through the AVERAGE center.

Using Data Points

When calculating slope, don't force yourself to pick original data points. They might not be on the line! Pick clear intersections on the grid that the LINE touches.

Real-Life Applications

Machine Learning:

Simple AI models use this exact same math.
It's called "Linear Regression." The computer takes millions of dots, finds the best line, and uses it to predict stock prices, weather, or user behavior.