Where we left off

Same data. Three lines. Only one of them is doing the job.

What is a best-fit line for?

A best-fit line answers two questions about a data set:

1. Is there a relationship between these two variables, and what shape is it?
2. What would happen at a value I didn't measure?

Neither question is about any single data point. Both are about the trend.

That is the whole idea: the line models the pattern, not the measurements.

Rule 1: do not connect the dots

Connecting the dots says "every wiggle is real."

But the wiggles come from your ruler, your parallax, your timing — not from the spring.

A best-fit line smooths through the uncertainty instead of tracing it.

Rule 2: the line does not have to touch anything

It is completely normal for a best-fit line to miss every single point.

What matters instead:

• roughly equal numbers of points above and below
• points evenly spread along the whole line
• the line stays close to the data overall

Before you draw: look

Ask yourself three things first.

• Does this look like a line, or a blob, or a curve?
• Is the trend positive (rises to the right) or negative (falls to the right)?
• Blur your eyes. A thick fuzzy band usually appears — that band is your line.

Your finished line should agree with what you just saw.

Method 1 — Area · Step 1

Draw a smooth shape that encloses all of the data.

Keep it simple and even — one closed loop around the whole cloud.

No spikes to reach a stray point.

Method 1 — Area · Step 2

Cut the shape in half with a straight line.

Edge to edge, splitting the enclosed area into two equal parts.

Half the area above, half below.

Method 1 — Area · Step 3

Erase the shape. Keep the line.

That is your best-fit line.

Notice it passes through none of the data points — and that is fine.

The shape tells you about your data

A fat blob means noisy data and a less certain line.
A skinny blob means precise data and a line you can trust further.

Method 2 — Dividing · Step 1

Split the data into two equal groups.

Draw a dashed vertical line with the same number of points on each side.

Odd number of points? Do your best — aim for half on each side.

Method 2 — Dividing · Step 2

Mark the center of each group with an ✕.

Eyeball the middle of each cluster — the balance point, not a data point.

Your partner's ✕ will land somewhere slightly different. Expected.

Method 2 — Dividing · Step 3

Connect the two ✕ marks and extend to the edges.

Done.

Both methods on this data set gave a slope near $0.21$ N/cm. That agreement is the point — good methods converge.

Check your line before you use it

The line on the left has about half the points above and below — but they are not evenly spread: everything low is under it, everything high is over it. That is a slope problem.

Why we care: the slope is physics

For this spring, $F = kx$, so

$$k = \frac{\Delta F}{\Delta x} \approx 0.21 \ \frac{\text{N}}{\text{cm}} = 21 \ \frac{\text{N}}{\text{m}}$$

Your slope is the spring constant. A sloppy line is a wrong measurement of $k$.

Take $\Delta F$ and $\Delta x$ from two points on the line, far apart — never from two data points.

Why we care: prediction

Stretch of $15$ cm was never measured. The line still answers it: $F \approx 3.1$ N.

• Interpolation (inside your data) — trustworthy
• Extrapolation (outside your data) — risky; the physics may change