Correlation

You’ll be able to

Interpret the correlation coefficient r as a measure of linear association
State the properties and limitations of correlation
Explain why a strong correlation does not confirm a linear relationship

What r measures

The correlation coefficient r measures the direction and strength of a linear association between two quantitative variables. It ranges from −1 to +1: values near +1 mean a strong positive linear pattern, values near −1 a strong negative linear pattern, and values near 0 little or no linear association. The sign of r always matches the direction (slope) of the pattern.

Properties and traps

Correlation has no units and does not change if you swap x and y or rescale the variables (converting inches to centimeters leaves r unchanged). But r has real limits: it only describes linear patterns — a perfect U-shaped curve can have r ≈ 0. It is not resistant: a single outlier can inflate or deflate it dramatically. And a large r never proves causation.

Correlation coefficient

r = (1/(n−1)) · Σ [ (x_i − x-bar)/s_x ] · [ (y_i − y-bar)/s_y ]

r is the average product of the z-scores of x and y. Because it uses standardized values, r has no units and is unaffected by changes of scale.

Worked example

A study reports r = 0.9 between hours of exercise per week and a fitness score. A classmate says "r = 0.9 means 90% of people get fit from exercise." Explain what r = 0.9 actually tells us.

1.r measures the direction and strength of a linear relationship, not a percentage of people.
2.The sign (+) means the association is positive: more exercise goes with higher fitness scores.
3.The magnitude (0.9, close to 1) means the linear association is strong — points cluster tightly around a line.
4.r says nothing about causation or about "90% of people."

Answer: r = 0.9 means there is a strong, positive linear association between weekly exercise and fitness score — as exercise increases, fitness score tends to increase, with points falling close to a straight line. It is not a percentage of people and does not prove exercise causes fitness.

Checkpoint

A scatterplot shows points forming a clear, symmetric U-shape (a strong curved pattern). What is the most likely value of r?

Watch out

A correlation near 0 does not mean the variables are unrelated — it means there is no strong linear relationship. Always look at the scatterplot: a curved pattern can be strong yet produce r ≈ 0. Never report r without confirming the form is linear.

Checkpoint

Heights and weights of adults are recorded with r = 0.72. The heights are then reconverted from inches to centimeters. What happens to r?

Answer the 2 checkpoints as you read.