Key takeaway: The correlation coefficient r is a number from −1 to 1 that tells the direction and strength of the linear relationship between two quantitative variables.
Grade 9 – Algebra 1S-ID.C.8Statistics and Data
Definition of Correlation Coefficient
The correlation coefficient, written \(r\), puts the look of a scatter plot into one number. The sign says the direction: positive when the points rise from left to right, negative when they fall. The size, without the sign, says the strength: close to 1 means the points hug a straight line, and close to 0 means there is almost no linear pattern.
A value of exactly 1 or −1 means all points lie on a line. The number only describes linear patterns, so a perfect curve such as a parabola can have \(r\) near 0. Correlation also cannot show cause and effect. Calculators and spreadsheets produce \(r\) from the data pairs; students are expected to interpret it rather than compute it by hand.
Formula and notation
The value always satisfies
\[-1 \le r \le 1\]
It has no units. Rough guide: \(|r| \ge 0.8\) is strong, about 0.5 is moderate and \(|r| \le 0.3\) is weak. The square \(r^{2}\) gives the share of the variation in \(y\) that the line explains.
Key facts
- The sign of r gives the direction; its absolute value gives the strength.
- Swapping the x and y variables leaves r unchanged.
- Changing units, such as feet to inches, does not change r.
- A strong correlation coefficient does not prove that one variable causes the other.
Correlation Coefficient example
A cafe compares daily high temperature with cups of hot cocoa sold and finds \(r = -0.82\). A second data set, rainfall and umbrella rentals, has \(r = 0.45\). Describe each and decide which linear relationship is stronger.
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\(r = -0.82\): negative, so cocoa sales fall as temperature rises. The size 0.82 is large, so the trend is strong.
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\(r = 0.45\): positive, so rentals rise with rainfall, but 0.45 is only moderate.
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Compare absolute values: \(|-0.82| = 0.82 > 0.45\).
The cocoa data show the stronger relationship, strong and negative; the umbrella data are moderate and positive.
Common mistake
Watch out
Hal says \(r = 0.45\) is a stronger relationship than \(r = -0.82\) because 0.45 is greater than \(-0.82\).
Why it is wrong
Strength depends on the distance from 0, not on the sign. The minus sign only means the trend goes downhill. Since 0.82 is bigger than 0.45, \(r = -0.82\) describes the tighter linear pattern. Compare the absolute values of the two coefficients.
Related terms and lessons
- Algebra 1 · Two-Way Tables, Scatter Plots and CorrelationLessonWorksheetQuiz
- AP Statistics · Regression AnalysisLessonWorksheetQuiz
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