Line of Best Fit Examples in Statistics

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Line of Best Fit.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Statistics.

Concept Recap

The line of best fit (trend line) is the straight line that best represents the overall trend in a scatter plot by minimizing the sum of squared vertical distances between the line and all data points. Its equation enables predictions for new x-values.

If you stretched a rubber band through a scatter plot to be as close to all points as possible, that's the line of best fit. It captures the overall trend.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Line of Best Fit asks whether the same cases connect two variables or groups in a pattern that can be described carefully.

Common stuck point: Students often know a procedure related to line of best fit but skip the recognition step: Am I studying a relationship between variables, and have I separated association from causation? That leads to a calculation or graph that looks reasonable but answers a different question.

Sense of Study hint: Ask: Am I studying a relationship between variables, and have I separated association from causation?

Worked Examples

Example 1

medium

Mean point is

(\bar{x},\bar{y}) = (4, 11)

and slope is

-2

. Write the line of best fit.

Answer

\hat{y} = -2x + 19

First step

Use

\hat{y} = mx + b

and the mean point.

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Example 2

medium

Trend line for study hours

x

vs. test score

\hat{y}

\hat{y} = 8x + 50

. Interpret the slope in context.

Example 3

medium

A scatter plot of weight (lb) vs. age (yr) for puppies gives

\hat{y} = 5x + 2

. Predict the weight at age

6

years and explain whether you should trust the prediction.

Example 4

hard

Data:

(1,2),(2,3),(3,5),(4,4),(5,6)

. Use

\bar{x}=3,\bar{y}=4

and slope formula

b = \frac{\sum(x_i-\bar{x})(y_i-\bar{y})}{\sum(x_i-\bar{x})^2}

to find the line.

Five data points with the computed line of best fit

Example 5

easy

A scatter plot shows the relationship between hours studied (x) and test score (y). The data points generally trend upward. A line of best fit has equation

y = 5x + 40

. (a) Interpret the slope. (b) Predict the score for a student who studies 8 hours.

Example 6

medium

Given five data points: (1,3), (2,5), (3,6), (4,8), (5,11). Estimate the line of best fit by finding the slope using the first and last points, then adjust to pass through the centroid

(\bar{x}, \bar{y})

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A line of best fit has equation

\hat{y} = 2x + 5

. Predict

\hat{y}

when

x = 3

Read off the predicted value at x = 3

Example 2

easy

A line of best fit is

\hat{y} = -x + 10

. Find the predicted value at

x = 4

Predict ŷ at x = 4

Example 3

easy

The line of best fit

\hat{y} = 3x + 1

has what slope?

Example 4

easy

The line of best fit

\hat{y} = 4x - 7

has what y-intercept?

Example 5

easy

A trend line is

\hat{y} = 0.5x + 2

. Predict

\hat{y}

x = 10

Predict ŷ at x = 10

Example 6

easy

As x increases, the line of best fit

\hat{y} = 2x + 3

predicts y to do what?

Example 7

easy

The line of best fit minimizes what quantity?

Example 8

easy

Trend line

\hat{y} = -2x + 20

. Predict

\hat{y}

x = 5

Predict ŷ at x = 5

Example 9

medium

Two points on a line of best fit are

(0, 4)

and

(2, 10)

. Find its equation.

Find the line through these two points

Example 10

medium

A line of best fit passes through the mean point

(\bar{x}, \bar{y}) = (5, 12)

with slope

2

. Find its equation.

Line of best fit passing through the mean point (5, 12)

Example 11

medium

Trend line

\hat{y} = 1.5x + 4

. By how much does

\hat{y}

change when x increases by 6?

How much does ŷ rise when x increases by 6?

Example 12

medium

A line of best fit predicts

\hat{y}=24

x=8

and

\hat{y}=30

x=12

. Find the slope.

Find the slope of the line through (8, 24) and (12, 30)

Example 13

medium

The line of best fit is

\hat{y}=2x+5

. At

x=4

the observed value is 15. Is the prediction an over- or under-estimate?

Is the observed point above or below the line?

Example 14

medium

A trend line

\hat{y}=-3x+40

models temperature drop. Predict

\hat{y}

x=7

and state the trend direction.

Example 15

medium

Why is fitting a line of best fit inappropriate for a clearly U-shaped (curved) scatter plot?

Example 16

medium

Line of best fit

\hat{y}=5x+3

is built from data with x ranging 1 to 10. Why is predicting at

x=50

risky?

Example 17

medium

A line of best fit

\hat{y}=2x+1

is fitted, but one extreme outlier at

(10, 100)

pulls the line up. What problem does this illustrate?

Example 18

challenge

A line of best fit passes through

(\bar{x},\bar{y})=(4,10)

and has slope

b = r\frac{s_y}{s_x}

with

r=0.8

s_y=6

s_x=4

. Find the full equation

\hat{y}=bx+a

Example 19

challenge

Two lines are proposed for data

(1,2),(2,2),(3,5)

: line A

\hat{y}=x

and line B

\hat{y}=1.5x-0.5

. Which has the smaller sum of squared residuals?

Example 20

challenge

A line of best fit

\hat{y}=mx+b

gives

\hat{y}=14

x=2

and

\hat{y}=26

x=5

. Predict

\hat{y}

x=8

Extend the line to predict at x = 8

Example 21

easy

A trend line is

\hat{y} = 0.8x + 4

. Predict

\hat{y}

when

x = 5

Predict ŷ at x = 5

Example 22

easy

Identify the slope of the line of best fit

\hat{y} = -1.5x + 12

Example 23

easy

Trend line

\hat{y} = 6 - 0.5x

. Predict

\hat{y}

x = 8

Predict ŷ at x = 8

Example 24

easy

What is the y-intercept of

\hat{y} = 7x - 15

Example 25

easy

Trend line

\hat{y} = 5x

. Predict

\hat{y}

x = 3

Example 26

medium

A trend line passes through

(1, 5)

and

(6, 20)

. Find its equation.

Find the equation of the line through these two points

Example 27

medium

Trend line

\hat{y} = 0.4x + 8

. By how much does

\hat{y}

change when

x

increases by

25

How much does ŷ change when x increases by 25?

Example 28

medium

A trend line gives

\hat{y} = 18

x = 4

and

\hat{y} = 6

x = 10

. Find the slope.

Find the slope between (4, 18) and (10, 6)

Example 29

medium

A trend line crosses the y-axis at

14

and has slope

-3

. Write the equation.

Line with slope −3 and y-intercept 14

Example 30

medium

Trend line

\hat{y} = 1.2x + 5

models price vs. demand. Which value of

x

gives predicted

\hat{y} = 23

For which x does the line predict ŷ = 23?

Example 31

medium

Trend line

\hat{y} = 2x - 5

. What is

\hat{y}

x = 0

Find ŷ at x = 0 (the y-intercept)

Example 32

hard

A regression line uses summary stats

\bar{x}=10

\bar{y}=40

r=0.5

s_y=12

s_x=4

. Find the line of best fit.

Regression line passing through the mean point (10, 40)

Example 33

hard

Why does the least-squares line minimize

\sum (y_i - \hat{y}_i)^2

rather than

\sum (y_i - \hat{y}_i)

Example 34

hard

Adding a single outlier far from the rest can change the line of best fit a lot. What property of least-squares causes this sensitivity?

Example 35

hard

A line of best fit has slope

0

and intercept

\bar{y}

. What does this say about the linear relationship between

x

and

y

Example 36

hard

Trend line

\hat{y} = 1.4x + 6

was fit on data with

x

between

2

and

20

. Should you use it to predict at

x = 200

? Why or why not?

Example 37

challenge

For the four points

(1,1),(2,3),(3,2),(4,4)

, find the line of best fit.

Find the line of best fit for these four points

Example 38

medium

A line of best fit for temperature (°C,

x

) vs ice-cream sales (units,

y

) is

y = 12x - 50

. (a) Predict sales when it is 30°C. (b) Below what temperature does the model predict zero or negative sales? (c) Comment on the limitation.

Example 39

hard

Two students draw different lines of best fit through the same scatter plot. Student A's line:

y = 3x + 2

(sum of squared residuals = 40). Student B's line:

y = 2.5x + 4

(sum of squared residuals = 28). Which line is better and why?

← Back to Line of Best Fit Formula Explained Practice Mode

Related Concepts

Scatter Plot Linear Regression Correlation Coefficient

Background Knowledge

These ideas may be useful before you work through the harder examples.

stat scatter plot