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What is Correlation?

Correlation measures the direction and strength of the linear relationship between two variables. The correlation coefficient r ranges from -1 to +1 and shows how closely two variables move together. It's closely related to regression, but a strong correlation never proves causation.

Short answer

Correlation is a coefficient, r = Cov(x,y) / (σx·σy), that measures the direction (positive or negative) and strength (from -1 to 1) of the linear relationship between two variables.

Positive vs. negative correlation

Positive correlation (r > 0)

•As x increases, y increases
•Scatter points slope upward
•Example: height & weight

Negative correlation (r < 0)

•As x increases, y decreases
•Scatter points slope downward
•Example: price & demand

Try it: interactive calculator

Correlation coefficient r

0.5

= 10/(4*5)

Covariance (x,y)10 Standard deviation of x4 Standard deviation of y5

Step-by-step worked examples

Find the correlation coefficient for x = (1, 2, 3) and y = (2, 4, 6).

x̄ = 2, ȳ = 4
Deviations x: -1, 0, 1 — y: -2, 0, 2
Σ(x-x̄)(y-ȳ) = 2 + 0 + 2 = 4
Σ(x-x̄)² = 2, Σ(y-ȳ)² = 8
r = 4 / √(2×8) = 4/4 = 1

Find the correlation coefficient for x = (1, 2, 3) and y = (6, 4, 2).

x̄ = 2, ȳ = 4
Deviations x: -1, 0, 1 — y: 2, 0, -2
Σ(x-x̄)(y-ȳ) = -2 + 0 - 2 = -4
Σ(x-x̄)² = 2, Σ(y-ȳ)² = 8
r = -4 / √(2×8) = -4/4 = -1

Find the correlation coefficient for x = (1, 2, 3, 4) and y = (2, 3, 5, 4).

x̄ = 2.5, ȳ = 3.5
Deviations x: -1.5, -0.5, 0.5, 1.5 — y: -1.5, -0.5, 1.5, 0.5
Σ(x-x̄)(y-ȳ) = 2.25+0.25+0.75+0.75 = 4
Σ(x-x̄)² = 5, Σ(y-ȳ)² = 5
r = 4 / √(5×5) = 4/5 = 0.8

Flashcards

Quick quiz

Q1.What does a correlation coefficient of r = -1 mean?

Correct answer: C. r = -1 means a perfect negative relationship — as x rises, y falls in perfect lockstep.

Q2.The range of the correlation coefficient r is:

Correct answer: B. r is always between -1 and +1.

Q3.Ice cream sales and drowning deaths show a high correlation. This means:

Correct answer: B. This is a classic confounding-variable example — there's no causal link.

Q4.If r = 0.9, the relationship is:

Correct answer: B. 0.9 is close to 1, indicating a strong positive relationship.

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Common mistakes

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Believing correlation proves causation. — Correct: Correlation only shows association — a lurking variable or coincidence may explain it.

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Thinking r = 0 means no relationship at all. — Correct: r = 0 only rules out a LINEAR relationship; a curved one can still exist.

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Confusing correlation strength with the regression slope. — Correct: r measures relationship strength (unitless); slope b measures rate of change (has units).

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Assuming r close to 0.5 is 'strong'. — Correct: Strength interpretation depends on the field, but generally |r| > 0.7 is considered strong.

FAQ

What is correlation?

Correlation is a statistical measure of the direction and strength of the linear relationship between two variables, ranging from -1 to +1.

What is the formula for correlation?

The formula is r = Cov(x,y) / (σx·σy) — covariance divided by the product of the two standard deviations, giving a unitless coefficient.

How do you calculate correlation?

Multiply each variable's deviation from its mean, sum those products, then divide by the square root of the product of the summed squared deviations.

What are examples of correlation?

Height and weight show positive correlation, price and demand often show negative correlation, and shoe size vs. reading ability shows a spurious correlation driven by age.