Scatter Plots & Association — I can create a scatter plot for bivariate data and describe association (direction, form, strength). - Pre Algebra

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.

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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.

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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.

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.

The three features of association

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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:

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

Exercise and heart rate data

Creating and Describing the Scatter Plot

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Step 1 — Set Up the Axes

The 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.
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Step 2 — Plot Each Ordered Pair

Write 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.
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Step 3 — Describe the Direction

As exercise days increase (move right), heart rate decreases (dots go down). This is a negative association.

Direction: negative
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Step 4 — Describe the Form

The dots follow a straight-line path, not a curve. This is a linear form.

Form: linear
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Step 5 — Describe the Strength

The dots sit very close to an imaginary straight line. There is very little scatter. This is a strong association.

Strength: strong
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Step 6 — Write a Full Description

Put 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.

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.

Pros and cons of scatter plots

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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.

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

From scatter plots to regression

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

Question 1: CONCEPTUAL

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

Show answer

A negative association means that as the values on the x-axis increase, the values on the y-axis decrease. On the scatter plot, the dots trend downward from left to right. For example, more hours of TV watching might be associated with lower test scores. The key word is 'decrease' — one variable goes up while the other goes down.

Question 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.

Show answer

These points lie exactly on a straight line going upward. The direction is positive because as x increases, y increases. The form is linear because the dots make a perfectly straight line. The strength is strong (in fact, perfect) because every dot sits right on the line with no scatter at all. You would say: 'There is a strong, positive, linear association.'

Question 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?

Show answer

The direction is negative because as the car gets older, its value drops. The form is roughly linear — most dots follow a straight-line trend. The strength is strong because the dots stay fairly close to an imaginary line. However, the last point (6, 8) drops more sharply than you would expect from the pattern. It could be considered a mild outlier or could suggest the relationship becomes nonlinear for very old cars. A good description would be: 'There is a strong, negative, approximately linear association with a slightly steeper drop at 6 years.'

Question 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.

Show answer

The association is negative and linear with moderate strength. A good real-world sentence would be: 'As the daily high temperature increases, the number of hot chocolates sold tends to decrease, following a roughly straight-line pattern with moderate scatter.' This makes sense because people are less likely to buy a hot drink on a warm day. The moderate strength tells us temperature is not the only factor — other things like time of year or personal preference also play a role.

Question 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.'

Show answer

The student is confusing association with causation. Bigger feet do not cause better reading. The hidden third variable is age. As children get older, their feet grow AND their reading skills improve. Age is the real driver of both changes. This is a classic example of why we say 'correlation does not equal causation.' Whenever you see a strong association, always ask whether a lurking variable could be responsible for the pattern in both variables.

SUMMARY

Lesson Summary

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.