Topic 2.1 Notes – Introducing Statistics: Are Variables Related?

What It Means for Two Variables to Be Related

Here we’re looking at bivariate data. That just means we measure two variables on the same individuals.

Example:

• For each student: hours of sleep and quiz score.
• For each voter: political party and opinion on a policy.

The big question is always:

If I know one variable, does that tell me something about the other?

That’s an association.

• If knowing one variable changes what we expect for the other → association.
• If the distribution of one variable looks the same no matter what the other variable is → no association.

Remember from Unit 1 that variation is everywhere. Even if two variables aren’t related, random chance can make small samples look patterned. That uncertainty is the heart of statistics.

Questions that signal this topic:

• Are these variables associated?
• Do the groups differ?
• Is there a trend as one variable increases?
• Could the pattern just be random?

Types of Bivariate Data

Everything depends on what type of variables you have.

1. Two Categorical Variables

Both variables put individuals into categories.

Examples:

• School type (public/private) and participation in sports (yes/no)
• Favorite subject and grade level

We compare conditional proportions.

Instead of asking “How many?”, we ask:

Of this group, what percent fall into each category?

If the conditional percentages are similar across groups → little/no association.
If they’re noticeably different → possible association.

Raw counts can mislead you. If one group is larger, it will naturally have larger counts. Percentages let you compare fairly.

2. Two Quantitative Variables

Both variables are numerical.

Examples:

• Minutes studied and exam score
• Outside temperature and ice cream sales

We use a scatterplot to look for a pattern.

We describe association using four things:

• Direction
  • Positive → as x increases, y tends to increase
  • Negative → as x increases, y tends to decrease
• Form
  • Linear or curved
• Strength
  • How tightly points follow a pattern
• Outliers
  • Unusual points that don’t fit the trend

Graphs Used to Detect Association

For Two Categorical Variables

A segmented bar graph helps compare conditional percentages. Each bar represents a group, and the segments within a bar show the conditional distribution for that group.

Segmented bar graph comparing conditional percentages

In the example above, each sport has a bar that totals 100%, and the colored segments show the percent in Track and Field and Swimming within that sport.

If the segments line up closely across groups, there’s likely no association.
If the segment sizes differ clearly, that suggests association.

A mosaic plot shows the same idea using area.

Mosaic plot comparing two categorical variables

Here, the widths of the columns represent the group sizes, and the colored areas show the conditional distribution within each group.

You’re always comparing conditional distributions.

For Two Quantitative Variables

The scatterplot is your main tool.

Scatterplot showing positive curved association

In this example, as study time increases, exam scores tend to increase, which indicates a positive association.

When you describe it, be specific:
“There is a strong positive curved association between minutes studied and exam score.”

That kind of wording shows you understand direction, form, and strength.

What Association Does and Does Not Mean

What it means:

• There is a pattern in the sample.
• Knowing one variable helps predict the other.

What it does not mean:

• It does not prove causation.
• It does not guarantee the pattern isn’t due to random variation.
• It does not mean the relationship holds for the whole population without further analysis.

Students lose points by jumping from “associated” to “causes.” Keep those separate.

Identifying the Statistical Question

When you see a scenario, pause and ask:

1. What are the two variables?
2. Are they categorical or quantitative?
3. What graph fits?
4. Are we investigating whether there’s an association?

On free-response questions, you may only be asked to identify the relationship type. On multiple choice, they love giving a graph and asking whether the evidence suggests association or could be due to random chance.