Skip to main content

​
Learning outcomes

  • understand the idea of covariance
  • interpret positive, negative, and near-zero covariance
  • distinguish covariance from correlation
  • avoid overinterpreting the magnitude of covariance

​
What is covariance?

  • Covariance measures how two numerical variables vary together.
Basic idea:
  • if large x values tend to occur with large y values, covariance is positive
  • if large x values tend to occur with small y values, covariance is negative

​
Interpretation of sign

  • positive covariance -> variables tend to move in same direction
  • negative covariance -> variables tend to move in opposite directions
  • near-zero covariance -> little linear joint movement

​
Why magnitude is hard to compare

  • Covariance depends on units.
  • Changing units can change the covariance value.
Example:
  • using centimeters vs meters changes the scale

​
Covariance vs correlation

  • covariance gives direction of joint movement
  • correlation standardizes the relationship to a scale between -1 and 1

​
Exam hints and traps

  • Covariance sign tells direction, not exact strength by itself.
  • A larger covariance is not always “stronger” if units differ.
  • Zero covariance does not automatically mean absolute independence.

​
Quick practice

  1. If x and y rise together, what is the covariance sign?
  2. Why is covariance hard to compare across datasets?
  3. Which is standardized: covariance or correlation?

​
Answer key

  1. Positive
  2. Because it depends on units and scale
  3. Correlation