Scatter Plots & Association — I can create a scatter plot...

SECTION 1

Historical Context & Motivation

Have you ever wondered if studying more hours actually leads to better test scores? Or if taller people really do weigh more? People have been asking questions like these for centuries. A scatter plot is a graph that helps us see the answer by plotting two pieces of information at the same time.

Long before computers, scientists and mathematicians needed a way to look at data and find patterns. They discovered that drawing dots on a grid was one of the best tools for the job. Let's look at how this idea developed over time.

1686

Early Data Graphs

Edmund Halley plotted data about air pressure and altitude. This was one of the first times someone used a graph to show how two measurements relate to each other.

1833

Scatter Plots Take Shape

English scientist John Herschel created some of the first true scatter plots. He placed individual dots on a grid to study orbits of stars.

1886

Studying Human Traits

Francis Galton used scatter plots to compare the heights of parents and their children. He found that the data formed interesting patterns.

Today

Scatter Plots Everywhere

Scientists, doctors, coaches, and business leaders all use scatter plots daily. They appear in news articles, school projects, and apps on your phone.

The big question scatter plots help us answer is: When one thing changes, does another thing change along with it? That idea is at the heart of this lesson.

SECTION 2

Core Principles & Definitions

Before we start drawing, let's nail down some key vocabulary. Each idea below is a building block you will use for the rest of this lesson.

Bivariate Data

Bivariate data means data with two variables (two pieces of information) collected about the same subject. Example: each student's study hours and test score.

Scatter Plot

A scatter plot is a graph that uses dots to show bivariate data. One variable goes on the x-axis and the other on the y-axis.

Association (Direction)

Direction tells us whether the dots go up or down as we move to the right. Positive means up; negative means down.

Association (Form)

Form describes the shape of the pattern. It can be linear (straight-line trend) or nonlinear (curved trend).

Association (Strength)

Strength tells us how tightly the dots cluster around a line or curve. Strong means tight; weak means spread out.

✦ KEY TAKEAWAY

Think of a scatter plot like a photo of a crowd. If the crowd is marching in one direction in a tight line, that's a strong, linear, positive association. If people are wandering all over with no pattern, there is no association. Your job is to describe the crowd!

SECTION 3

Visual Explanation — Building a Scatter Plot

Let's see what a scatter plot actually looks like. Imagine you asked eight friends how many hours they studied for a math test and then recorded their scores. The diagram below shows how to turn that table of numbers into a scatter plot.

Each cyan dot represents one student. The horizontal axis shows study hours, and the vertical axis shows the test score. Notice how the dots trend upward from left to right — that's a positive association.

To build this plot yourself, follow three steps. First, draw an x-axis and a y-axis and label them with your two variables. Second, choose a scale that fits all your numbers. Third, plot each pair of values as a single dot. That's it — you have a scatter plot!

SECTION 4

How to Plot Ordered Pairs

Each data pair in a scatter plot is written as an ordered pair (x, y). The first number tells you how far to go along the horizontal axis. The second number tells you how far to go up the vertical axis.

ORDERED PAIR FORMAT

(x, y)

x = value of the first variable (horizontal axis), y = value of the second variable (vertical axis). For example, (3, 75) means 3 study hours and a test score of 75.

When you pick which variable goes on each axis, there is a helpful rule. The variable you think might cause a change goes on the x-axis. The variable you think might respond to that change goes on the y-axis.

AXIS RULE

x-axis → explanatory variable, y-axis → response variable

Example: Study hours (explanatory) goes on the x-axis because you choose how long to study. Test score (response) goes on the y-axis because it responds to study time.

💡 Quick Tip

Always start each axis at a number that makes sense for your data. You don't have to start at zero if all your values are large. For example, if scores range from 55 to 95, you could start the y-axis at 50.

SECTION 5

Direction, Form & Strength — A Closer Look

Once you have your scatter plot, it's time to describe the pattern. You will use three words: direction, form, and strength. The diagram below shows six common patterns.

Six common patterns in scatter plots. Top row, left to right: strong positive linear, strong negative linear, and weak positive linear. Bottom row: nonlinear (curved), no association, and cluster pattern.

The three features of association

Feature	Options	What to Look For
Direction	Positive, Negative, or None	Do the dots rise (positive), fall (negative), or show no clear trend (none) as you move right?
Form	Linear or Nonlinear	Do the dots follow a straight-line path (linear) or a curved path (nonlinear)?
Strength	Strong, Moderate, or Weak	How close do the dots stay to the imaginary line or curve? Tight = strong, spread out = weak.

⚠️ Watch Out for Outliers!

An outlier is a dot that sits far away from the rest of the data. One or two outliers can make a pattern look weaker than it really is. Always mention outliers when you describe a scatter plot.

SECTION 6

Worked Example — From Data to Description

A gym teacher recorded the number of days each student exercised per week and their resting heart rate (beats per minute). Here is the data set:

Exercise and heart rate data

Student	Exercise Days / Week (x)	Resting Heart Rate (y)
A	1	85
B	2	80
C	2	82
D	3	76
E	4	72
F	5	68
G	6	64

Creating and Describing the Scatter Plot

Step 1 — Set Up the AxesThe explanatory variable is Exercise Days per Week, so it goes on the x-axis. The response variable is Resting Heart Rate, so it goes on the y-axis. Choose a scale: x from 0 to 7, y from 60 to 90.

Step 2 — Plot Each Ordered PairWrite each row as an ordered pair: (1, 85), (2, 80), (2, 82), (3, 76), (4, 72), (5, 68), (6, 64). Place a dot at each location on the grid.

Step 3 — Describe the DirectionAs exercise days increase (move right), heart rate decreases (dots go down). This is a negative association.

Direction: negative

Step 4 — Describe the FormThe dots follow a straight-line path, not a curve. This is a linear form.

Form: linear

Step 5 — Describe the StrengthThe dots sit very close to an imaginary straight line. There is very little scatter. This is a strong association.

Strength: strong

Step 6 — Write a Full DescriptionPut it all together in one sentence.

There is a strong, negative, linear association between exercise days per week and resting heart rate.

SECTION 7

Strengths & Limitations of Scatter Plots

Scatter plots are super useful, but they have limits. The table below shows what they do well and where you need to be careful.

Pros and cons of scatter plots

Strengths ✅	Limitations ⚠️
Show the relationship between two variables at a glance.	Can only display two variables at a time.
Make outliers easy to spot.	Association does NOT prove causation (that one thing causes the other).
Help you choose the right type of model (line vs. curve).	Hard to read when there are too many overlapping dots.
Work well for any size data set.	Choosing bad axis scales can make patterns look stronger or weaker than they are.

⚡ CORRELATION ≠ CAUSATION

Just because two things go up together does not mean one causes the other. Ice cream sales and sunburn cases both rise in summer, but eating ice cream doesn't give you a sunburn — hot weather causes both! Always ask yourself: could there be a hidden third factor?

SECTION 8

Connection to Advanced Ideas

Scatter plots are the first step on a bigger journey. In later math and science classes, you will use scatter plots as a starting point for more powerful tools. Here is a quick preview.

From scatter plots to regression

What You Learn Now	What Comes Next
Describe association as positive, negative, or none	Calculate correlation coefficient (r) — a number between −1 and 1 that measures strength and direction exactly
Describe form as linear or nonlinear	Write the equation of the best-fit line (linear regression)
Eyeball the trend	Use technology to find the line of best fit and make predictions
Note outliers	Analyze residuals (the distances between dots and the line) to check your model

For now, focus on the skills in the left column. Mastering them will make those advanced topics feel much easier when the time comes!

SECTION 9

Practice Problems

PROBLEM 1 — CONCEPTUAL

What does it mean when we say a scatter plot shows a 'negative association'?

PROBLEM 2 — BASIC CALCULATION

A student collected these ordered pairs: (1, 10), (2, 20), (3, 30), (4, 40). Plot these four points in your mind (or on paper). Describe the direction, form, and strength of the association.

PROBLEM 3 — INTERMEDIATE

Here is data about the age of a car (in years) and its value (in thousands of dollars): (1, 25), (2, 22), (3, 19), (4, 17), (5, 15), (6, 8). Describe the association. Does any point stand out?

PROBLEM 4 — APPLIED

A weather station records the daily high temperature (°F) and the number of hot chocolates sold at a café. The scatter plot shows dots that go from upper-left to lower-right with moderate spread. Write a real-world sentence that describes this association.

PROBLEM 5 — CRITICAL THINKING

A student finds a strong positive association between shoe size and reading level among children ages 5 to 18. She concludes that bigger feet help you read better. What is wrong with her reasoning? Explain using the idea of a 'hidden third variable.'

SUMMARY

Lesson Summary

A scatter plot displays bivariate data by plotting each pair of values as a dot on a coordinate grid. The explanatory variable goes on the x-axis, and the response variable goes on the y-axis. Once the dots are plotted, you describe the pattern using three features: direction (positive, negative, or none), form (linear or nonlinear), and strength (strong, moderate, or weak).

Always look for outliers — dots that don't fit the pattern. And remember: association does not prove causation. A hidden third variable could be driving both changes. Mastering scatter plots prepares you for linear regression and the correlation coefficient in future courses.