Algebra
Data and statistics
Correlation coefficient
20 practice questions
2 video lessons
Theory + worked examples
Theory
The correlation coefficient \(r\) measures a linear relationship:
\[-1\le r\le 1.\]
- The sign gives the direction (positive or negative).
- \(|r|\) near \(1\) is a strong fit; near \(0\) is weak.
\(r=\pm1\) means the points lie exactly on a line.
\(r\) runs from \(-1\) to \(1\).
Interpreting \(r\).
The range:
\[-1\le r\le 1\]
Sign is direction; magnitude is strength.
How to interpret r
- Read the sign for the direction.
- Read \(|r|\) for the strength.
- Near \(1\): strong; near \(0\): weak.
- Use technology to compute \(r\).
Example 1 — Strong positive
Describe a correlation with \(r=0.95\).
Solution
\(|r|\) near \(1\) and positive.
| \(r=0.95\) | \(\Rightarrow\) | \(\text{strong positive}\) |
Example 2 — Weak
Describe \(r=0.1\).
Solution
\(|r|\) near \(0\) is weak.
| \(r=0.1\) | \(\Rightarrow\) | \(\text{weak}\) |
Example 3 — Negative
Describe \(r=-0.8\).
Solution
Negative and fairly strong.
| \(r=-0.8\) | \(\Rightarrow\) | \(\text{strong negative}\) |
Example 4 — Range of r
What values can \(r\) take?
Solution
From \(-1\) to \(1\).
| \(-1\) | \(\le\) | \(r\le1\) |
Common pitfalls
\(|r|\) near 1 is strong, near 0 is weak.
\(r\) measures linear fit only.
A strong \(r\) is not causation.
Frequently asked questions
What does the correlation coefficient measure?
The direction and strength of a linear relationship.
What is the range of r?
\(-1\) to \(1\).
What does \(r=0.95\) mean?
A strong positive correlation.
What does \(r=0\) mean?
No linear correlation.
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