Computing and Interpreting the Correlation Coefficient

Lesson 1 of 1: A Number for Strength

In this lesson:

• Read r as direction and strength on −1 to +1
• Know what r cannot tell you

Learning Objectives for This Lesson

By the end of this lesson, you should be able to:

1. State that r measures linear strength and direction
2. Compute r using technology
3. Interpret r's sign and magnitude, matched to a plot
4. Recognize that r is not causation

The Plot Trends Up — But How Strongly?

A scatter plot trends upward. How strong is the relationship?

• A tight band hugging a line, or a loose drifting cloud?
• "Strong" and "weak" aren't measurements

We want one number for the tightness. That number is r.

Tight Band or Loose Cloud, Quantified

You already judge strength by eye: tight band vs loose cloud.

• A tight band → strong; a loose cloud → weak
• r puts a number on the tightness you already see

r quantifies your eyeball judgment. Meet its scale.

r Measures Strength and Direction, −1 to +1

Sign gives direction; distance from zero gives strength.

The Three Landmark Correlation Values

Exact upward line, exact downward line, no pattern at all.

Sign Is Direction; Magnitude Is Strength

Read r in two separate steps:

• Sign → direction (up or down), matches the slope's sign
• Magnitude → strength (closeness to a line)

is very strong (downward); is weak. r is not the slope.

Your Turn: Read Direction and Strength

State the direction and strength for each:

Read sign and magnitude separately, then advance.

Answer: 1. Up, strong. 2. Down, weak. 3. No linear relationship.

Now Read r and Picture the Cloud

Two skills make r meaningful with real data:

• Get r from technology
• Picture the cloud an r value implies

An r you can't picture is empty. Let's do both.

Get r From a Calculator or Spreadsheet

The standard expects technology — no by-hand formula:

• Calculator: LinReg with diagnostics on reports r
• Spreadsheet: CORREL returns r from two columns

Read r off the output; the interpreting is your job.

Calibrate: Match Plots to r Values

Notice r ≈ 0.5 is a loose cloud — moderate, not strong.

Estimate r From a Plot, Then Reveal

A scatter plot shows a fairly tight upward band.

• Estimate r: near 0.2, 0.5, or 0.9?
• Commit before advancing

A tight upward band means strong positive.

Answer: Near 0.9 — a tight upward band is a strong positive correlation.

Your Turn: Estimate and Compute

1. A loose upward cloud — estimate r and justify
2. How would you compute r in a spreadsheet?

Estimate one, state the method for the other, then advance.

Answer: 1. About 0.3–0.5 (loose → moderate). 2. CORREL on the two columns.

r Is Powerful, but It Has Blind Spots

Interpreting r honestly means knowing three limits:

• It sees only straight-line patterns
• It's sensitive to outliers
• A high r is not causation

Let's take each in turn — the last sets up the next lesson.

r Is Linear-Only: Curves Read Near Zero

• The variables are clearly related — but not linearly
• r near 0 means no linear pattern, not "unrelated"

r Is Sensitive to a Single Outlier

One off-trend point can swing r noticeably.

• r is computed from all points together
• An extreme point carries outsized weight

Scan the plot for an outlier before trusting r.

A High r Does Not Prove Causation

A strong correlation does not mean one variable causes the other.

• r measures that they move together, not why
• Cause requires far more (next lesson)

r measures association strength — never proof of cause.

Why: Ice-Cream Sales and Drowning Both Rise

Ice-cream sales and drowning deaths correlate strongly.

• Eating ice cream doesn't cause drowning
• Summer heat drives both — a hidden common cause

Real, strong correlation — but no causation. (Next: C.9)

Quick Check: The Limits of High r

More firefighters at a fire correlates with more damage (high r).

• Do firefighters cause the damage?
• What does the high r actually tell you?

Decide and explain, then advance.

Answer: No — bigger fires cause both; r shows association only, not cause.

Your Turn: Three Ways r Misleads

Address each limit:

1. Strong curve, yet r ≈ 0 — why?
2. One off-trend point — what about r?
3. "Screens cause poor sleep" — what's wrong?

Address all three, then advance.

Answer: 1. r is linear-only. 2. Recompute without it. 3. Correlation isn't cause; a lurking variable may explain it.

Full Task: Compute, Interpret, Match, Limit

Given a bivariate data set:

1. Compute r with technology
2. Interpret sign (direction) and magnitude (strength)
3. Match it to the plot; name one limit

Do all four, then advance.

Answer: Read r off the tool; state direction + strength; confirm vs the plot; cite linear-only, outlier, or not-cause.

Key Takeaways and What's Next

✓ r measures linear strength and direction

✓ Sign = direction; magnitude = strength

✓ r is unitless — not the slope

r is linear-only — a strong curve reads near 0

A high r is never proof of causation

Next: correlation versus causation (C.9).