Scatterplots & Line of Best Fit — Core Skills

Scatterplots & Line of Best Fit: The Basics

Part 1 of 2 — Read the Direction, Use the Equation

A scatterplot is a graph made of dots. Each dot is one thing that got measured twice — for example, one student's study hours and that same student's test score.

A line of best fit is a single straight line drawn through the middle of those dots. It is not meant to touch every dot. It is meant to show the general direction they are heading.

Direction: is it going up or down?

Read the dots from left to right, the same way you read a sentence.

• Dots rising from left to right → positive association. As $x$ gets bigger, $y$ tends to get bigger too.
• Dots falling from left to right → negative association. As $x$ gets bigger, $y$ tends to get smaller.
• Dots with no pattern at all → no association.

The equation of the line

The line of best fit comes with an equation that looks like this:

$y = 2x + 5$

Two numbers matter, and each one has a plain-English meaning.

• The number in front of $x$ is the slope. Here the slope is $2$. It means: every time $x$ goes up by $1$, $y$ goes up by about $2$.
• The number added at the end is the y-intercept. Here it is $5$. It means: when $x$ is $0$, the predicted $y$ is $5$.

A positive slope means the line goes up. A negative slope, like the $-3$ in $y = -3x + 40$, means the line goes down.

Worked example: making a prediction

Using $y = 2x + 5$, predict $y$ when $x = 4$.

Put $4$ in for $x$:

$y = 2(4) + 5$

Multiply first: $2 \times 4 = 8$.

$y = 8 + 5$

Add: $y = 13$.

The predicted value is $13$. Substitute, multiply, add. That is the whole move.

Real dots versus the line

Sometimes a real dot sits above or below the line. The gap between them is called the residual, and you find it by subtracting:

residual $=$ actual $-$ predicted

If the real value is $15$ and the line predicted $13$, the residual is $15 - 13 = 2$. A positive residual means the dot sits above the line.

Scatterplots & Line of Best Fit: Practice

Part 2 of 2 — Four Moves

Almost every core scatterplot question is one of these four moves.

1. Name the direction. Read the dots left to right. Rising is a positive association. Falling is a negative association. No pattern means no association.
2. Predict a value. Take the equation, put the given number in for $x$, multiply, then add. For $y = 4x + 3$ at $x = 2$: $4 \times 2 = 8$, then $8 + 3 = 11$.
3. Explain the slope. The slope is the number in front of $x$. It says how much $y$ changes each time $x$ goes up by $1$. A positive slope means $y$ goes up; a negative slope means $y$ goes down.
4. Explain the y-intercept. The y-intercept is the number added at the end. It is the predicted $y$ when $x = 0$ — the starting value.

If a question asks about one real dot

• A dot far away from the general pattern is called an outlier.
• The gap between a real dot and the line is the residual: actual $-$ predicted. Predict first, then subtract. Above the line gives a positive residual; below the line gives a negative one.

One thing to remember

The line of best fit gives a prediction, not a guarantee. Real dots sit a little above or below it, and that is normal.