Do two things move together? Covariance and correlation answer that, but only one is easy to compare.
Introduction
Do Variables Move Together?
Do two things move together? Statistics has two tools to answer that, and most people mix them up.
- Two tools
- Often confused
Covariance: Multiply Deviations
Start with covariance. For each point, measure how far x and y sit from their averages, then multiply.
- Distance from each mean
- Multiply the pair
Signs Tell the Story
Points in the upper right and lower left give positive products. The other quadrants give negative ones. Average them all.
- Positive: move together
- Negative: move opposite
The Scale Problem
But covariance has a flaw. Change the units, say meters to centimeters, and the number balloons. It's hard to compare.
- Depends on units
- Hard to compare
Correlation: Rescaled
Correlation fixes this. Divide covariance by both standard deviations. The result always lands between negative one and one.
- Divide by both std devs
- Always between -1 and 1
Seeing Correlation
Watch it change. Tight upward lines near one. A shapeless cloud near zero. Tight downward lines near negative one.
- Near 1: rising
- Near 0: no line
- Near -1: falling
Real-World Uses
Investors use covariance to build diversified portfolios. Data scientists use correlation to pick features and spot redundant variables.
- Portfolio risk
- Feature selection
- Science research
Correlation Is Not Causation
One warning: correlation is not causation. It shows a pattern, never the reason. Follow for more quick math.
- Patterns, not reasons