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Fit Lines to
Scatter Plots

Learn to draw a line of best fit through scatter plot data and judge how well the line models the relationship between two variables.

8th Grade Math

What Is a Scatter Plot?

A scatter plot is a graph that shows individual data points plotted on a coordinate plane. Each point represents a pair of values: one for the independent variable (\( x \)) and one for the dependent variable (\( y \)).

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Recognizing Linear Association

When the data points in a scatter plot cluster along a straight-line path, we say the two variables have a linear association. The points don't need to fall exactly on a line: they just need to follow a general straight trend.

Positive Association
As \( x \) increases, \( y \) also increases. The cluster tilts upward to the right.
Negative Association
As \( x \) increases, \( y \) decreases. The cluster tilts downward to the right.
No Association
Points are scattered randomly with no visible pattern or direction.

Why Fit a Straight Line?

Straight lines are the simplest model for describing a relationship. Once we have a line, we can:

1
Summarize the trend in the data with a single equation \( y = mx + b \).
2
Make predictions for values of \( x \) we haven't measured yet.
3
Identify outliers: points that fall far from the line and behave unusually.

The Line of Best Fit

A line of best fit is the straight line that comes closest to the most data points. Informally, it should split the data so that roughly half the points are above the line and half are below.

\( x \)
\( y \)
 line of best fit above below

The Equation: \( y = mx + b \)

Every straight line can be described by the slope-intercept form:

\( m \) : Slope
The rate of change: how much \( y \) changes for every 1-unit increase in \( x \). Computed as \( m = \dfrac{\Delta y}{\Delta x} \).
\( b \) : \( y \)-Intercept
The value of \( y \) when \( x = 0 \). It is where the line crosses the \( y \)-axis.

Strong vs. Weak Fit

After drawing a line, we judge how well it fits by looking at how close the data points are to the line. Here are two scatter plots with the same line drawn through them:

Strong Fit Weak Fit

How to Draw a Line of Best Fit

1
Plot all data points on the scatter plot carefully.
2
Look at the overall direction: does the data trend upward, downward, or show no pattern?
3
Place a ruler so that the line passes through the middle of the cluster, with roughly equal numbers of points on each side.
4
Minimize gaps: adjust the line so the distances from the points to the line are as small as possible overall.
5
Draw and extend the line across the graph. Read off two clear points to find \( m \) and \( b \).

Example 1: Finding the Line

A teacher records the number of absences (\( x \)) and the final grade (\( y \)) for 8 students. The scatter plot is shown below. Fit a line and find its equation.

Absences
Grade (%)
 0 2 4 6 8 10 12 50 60 70 80 90
(0, 95)
(12, 52)

Interpreting \( y = -3.6x + 95 \)

Slope \( m = -3.6 \)
For every additional absence, the predicted grade drops by about 3.6 points.
\( y \)-intercept \( b = 95 \)
A student with zero absences would be predicted to earn about a 95%.

Example 2: Positive Association

A store records the daily temperature (\( x \), in °F) and bottles of water sold (\( y \)). Fit a line to the data below.

Temperature (°F)
Bottles Sold
 60 70 80 90 100 0 20 40 60 80
(60, 5)
(100, 75)

Example 3: Judging the Fit

Two students each drew a line for the same data. Which line is a better fit?

Student A Student B

Checking Fit with Residuals

A residual is the vertical distance from a data point to the line. It tells you how far off the prediction is for that point:

Positive residual
The point is above the line. The actual value is higher than predicted.
Negative residual
The point is below the line. The actual value is lower than predicted.

When a Line Doesn't Fit

Not every scatter plot should be modeled with a straight line. If the data curves, a line will systematically miss the pattern:

bad fit

Common Mistakes

✕ Connecting the Dots
The line of best fit is one straight line, not a zigzag through every point.
✕ Forcing Through the Origin
The line does not have to pass through \( (0,0) \) unless the data supports it.
✕ Ignoring Outliers
A single extreme point can pull the line. Note outliers but don't let them dominate the fit.

Summary Table

Concept Key Detail
Scatter Plot Graph of data pairs Each dot is \( (x, y) \)
Linear Association Points cluster along a line Can be positive or negative
Line of Best Fit Straight line through the middle Balanced above & below
Slope \( m \) Rate of change \( m = \frac{\Delta y}{\Delta x} \)
\( y \)-Intercept \( b \) Value when \( x = 0 \) Starting point of the line
Residual Actual minus predicted Small residuals = good fit

Question 1 of 5

A scatter plot shows data points that cluster tightly along a line sloping downward to the right. This is an example of:

Question 2 of 5

A line of best fit passes through the points \( (2, 14) \) and \( (8, 32) \). What is the slope of this line?

Question 3 of 5

A student's line of best fit is \( y = 2x + 10 \). The actual data point at \( x = 5 \) is \( y = 18 \). What is the residual?

Question 4 of 5

Which of the following is the best description of a good line of best fit?

Question 5 of 5

A scatter plot shows data arranged in a clear U-shape. A student draws a straight line through the middle. What is wrong with this approach?