Fitting a Line to Data

You will be able to:

• Draw an informal line of best fit through a scatter plot
• Use the three criteria for a good fit
• Explain why the line is a model, not the data

Learning Objectives for This Lesson

By the end of this lesson, you will be able to:

1. Informally fit a straight line to a scatter plot
2. Apply the three criteria for a good fit
3. Explain that a fitted line is a model, not the data
4. Recognize several reasonable lines can fit one data set

Can One Line Capture This Cloud?

Sixteen students: study hours and test scores. Could one line stand in for all of it?

Everyone's Line Is a Little Different

Three students drew three lines. All upward, none identical — and all reasonable.

A Model Is a Deliberate Summary

• A linear model is a straight line we draw to capture a data trend
• We choose it to simplify the cloud into one relationship
• It answers: "given , what would I expect?"

The Line Is a Model, Not the Data

• Real data are messy — points scatter around the trend
• The line smooths out that noise
• Some points sit above it, some below, a few far off

Why Doesn't the Line Hit Every Point?

If the line is a good model, why does it miss most of the dots?

Think about what the line is for before the next slide.

Criterion 1: The Line Follows the Trend

• The line slopes the same direction as the data
• Upward for a positive association
• Downward for a negative association

Criterion 2: Balanced Above and Below

• Roughly equal points above and below
• Of 16 points: about 8 above, 8 below

Criterion 3: Close to the Points

• The points should sit near the line
• Small total distance from points to line
• Closeness is judged as a whole, not point by point

Many Lines Can Be Good

• There is no single exact line when we fit by eye
• Several lines can satisfy all three criteria
• "Best fit" here means reasonable, not unique

Fitting the Study-Hours Data Line

Observe the trend, position the line through the middle, check closeness, draw.

Verify the Balance by Counting

• Count the points above the line
• Count the points below the line
• Adjust the line until the counts are close

How Many Points Should Sit Above?

You drew a line through 16 points. Roughly how many should be above it?

Decide your number, then advance.

Must the Line Touch Data Points?

• Through-points line: hits two dots, ignores the rest
• Balanced line: touches none, fits all

Watch Out for Outliers Pulling the Line

• A line pulled to the extremes misses the bulk
• Count: 14 below, 2 above — not balanced
• Tilt toward the middle of the data

Which Criterion Does This Line Break?

A nearly flat line is drawn through the upward study-hours data.

• It does not follow the upward trend
• Which of the three criteria fails first?

Which of These Two Lines Fits Better?

Two lines are drawn on the same scatter plot:

• Line A follows the trend and balances the points
• Line B is steep and leaves most points below it

Pick the better fit and justify it with a criterion.

Your Turn: Draw and Defend

A fresh scatter plot is given with a clear positive association.

1. Draw your own line of best fit
2. Write which criteria you used to place it

Commit to your line alone first, then compare with a partner.

Three Common Line-Fitting Traps to Avoid

Through the points: the line models the trend, not individual dots
One exact line: several reasonable lines can all be good
Chasing outliers: balance to the middle, not the extremes

What a Fitted Line Really Is

✓ A model: a deliberate straight-line summary of a trend
✓ Good fit: follows the trend, balances points, stays close
✓ Several reasonable lines can fit the same data

Coming Up Next: Assessing and Using Lines

Next lesson, you will judge how good your line is, learn when a line should not be drawn at all, and turn the line into an equation you can compute with.