Work out covariance and correlation and explain what they show
The idea
Covariance measures whether two variables move together: positive if they tend to rise and fall in step, negative if they offset. Its magnitude depends on the units of both variables, so it cannot be compared across pairs. Correlation divides covariance by the product of the two standard deviations, producing a unitless number bounded between -1 and +1 that can be compared across pairs. Neither says anything about causation, and both capture only linear association — two variables can be strongly related and still show a correlation near zero if the relationship curves.
Formulas
Correlation from covariance
ρ(A,B) = Cov(A,B) ÷ (σA × σB)
- Cov(A,B)
- the covariance of the two returns
- σA
- the standard deviation of A
- σB
- the standard deviation of B
Correlation is covariance made unit-free, so it is bounded by −1 and +1. Covariance alone says direction but not strength.
A practice question
What is the range of a correlation coefficient?
- −1 to +1 — correct
- 0 to 1Why not: That excludes inverse relationships, which correlation expresses as negative values down to −1.
- UnboundedWhy not: That describes covariance, whose size depends on the units of both variables. Normalising by the two standard deviations imposes the bound.
Dividing covariance by both standard deviations normalises it onto a fixed scale from −1 to +1, which is what makes correlations comparable across different pairs of variables.
A harder one
Why is correlation usually reported rather than covariance?
- Covariance cannot be negativeWhy not: It can, and a negative covariance is how an inverse relationship shows up before normalising.
- Covariance depends on the units of both variables, so it cannot be compared across pairs — correct
- Correlation captures non-linear relationshipsWhy not: It captures only linear association — that is its central limitation, not its advantage.
A covariance of 0.0012 means nothing without knowing the scales involved. Correlation strips the units out, which is the only reason two pairs can be ranked against each other.
The hardest tier
Two variables have a correlation near zero but are clearly related when plotted. What explains this?
- The sample is too small for correlation to be definedWhy not: It is defined for any sample with variation in both variables, and the plot shows a clear pattern rather than a sparse one.
- Correlation must have been calculated incorrectlyWhy not: A correctly computed correlation can sit at zero for a strongly dependent pair, provided the dependence is not linear.
- Correlation measures only linear association, and the relationship is non-linear — correct
A U-shaped relationship has rising and falling segments that cancel, leaving a correlation near zero despite a strong dependence. Plotting the data is the standard defence against trusting the coefficient alone.
In the app: 12 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
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