Pearson Correlation
Measures linear correlation between two variables.
Formula
Variables
Example
r = +1 is a perfect straight-line rise, r = 0 means no linear relationship.
How it works
Pearson correlation quantifies how tightly two variables move together in a straight-line pattern, using a standardized version of covariance. The numerator adds up paired deviations from each mean, so values are positive when both variables are above or below their means together and negative when one is above while the other is below. The denominator rescales by the variables’ overall spread, which makes the result unitless and forces the value to stay between -1 and 1.
When to use it
Use this when you want to measure linear association between two quantitative variables, such as height vs. weight, study time vs. exam score, or temperature vs. electricity demand. It is especially useful when you need a comparable, scale-free measure of strength and direction.
Worked example, step by step
Data: (x, y) = (1,2), (2,3), (3,5), (4,4) Step 1: Find the means: x̄ = (1+2+3+4)/4 = 2.5, ȳ = (2+3+5+4)/4 = 3.5 Step 2: Compute deviations and products: (1-2.5)(2-3.5) = (-1.5)(-1.5) = 2.25 (2-2.5)(3-3.5) = (-0.5)(-0.5) = 0.25 (3-2.5)(5-3.5) = (0.5)(1.5) = 0.75 (4-2.5)(4-3.5) = (1.5)(0.5) = 0.75 Step 3: Sum the products: Σ(xi-x̄)(yi-ȳ) = 2.25+0.25+0.75+0.75 = 4.00 Step 4: Sum squared deviations for x: Σ(xi-x̄)^2 = 2.25+0.25+0.25+2.25 = 5.00 Step 5: Sum squared deviations for y: Σ(yi-ȳ)^2 = 2.25+0.25+2.25+0.25 = 5.00 Step 6: Substitute into the formula: r = 4.00 / sqrt(5.00×5.00) = 4.00 / 5.00 = 0.80 Final answer: r = 0.80, which indicates a fairly strong positive linear relationship.
Common mistakes to avoid
- Interpreting r as causation instead of just association
- Using Pearson correlation for clearly curved or non-linear patterns
- Forgetting that outliers can strongly distort the result
Did You Know?
Correlation does not imply causation — ice-cream sales and drownings rise together, but neither causes the other.
Share this formula
More in Statistics & Probability
View allArithmetic Mean
BasicThe average of a data set.
Median
BasicThe middle value separating the higher and lower halves.
Variance
IntermediateAverage squared deviation from the mean.
Standard Deviation
IntermediateSquare root of the variance; typical spread.
Z-Score
IntermediateNumber of standard deviations a value is from the mean.
Bayes' Theorem
IntermediateUpdates a probability given new evidence.