Probability course
Probability course
Module 6 of 8

Variance & covariance

Track spread, scale, and the way variables move together.

18 min2 worked examples3 concept checks

Loading your learning progress…

Learning objectives
  • Calculate variance using a second moment.
  • Handle translations and scaling correctly.
  • Include covariance when adding dependent quantities.

Concepts

Spread around a mean

Variance is Var⁡(X)=E[(X−E[X])2]=E[X2]−E[X]2\operatorname{Var}(X)=E[(X-E[X])^2]=E[X^2]-E[X]^2 for finite second moments. It uses squared units. Standard deviation is the nonnegative square root and has the original units. Two distributions can share a mean while carrying very different uncertainty.

Shifting does not spread; scaling does

Adding a constant moves every outcome and the mean equally, leaving deviations unchanged. Multiplying by a changes each deviation by a and its square by a²: Var⁡(aX+b)=a2Var⁡(X)\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X). A negative multiplier also squares, so variance cannot become negative.

The missing cross-term

Var⁡(X+Y)=Var⁡(X)+Var⁡(Y)+2Cov⁡(X,Y)\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y)+2\operatorname{Cov}(X,Y). Covariance records whether deviations tend to align. Independence implies zero covariance when second moments exist, but zero covariance does not generally imply independence. For a weighted sum, the cross-term becomes 2ab Cov(X,Y).

Examples

Worked example 1

Spread of a fair die

Find the variance of a fair six-sided die value X.

  1. Find the mean

    E[X]=(1+2+3+4+5+6)/6=7/2.

  2. Find the second moment

    E[X²]=(1+4+9+16+25+36)/6=91/6. Square the outcomes before averaging.

  3. Subtract the squared mean

    Var⁡(X)=91/6−(7/2)2=182/12−147/12=35/12\operatorname{Var}(X)=91/6-(7/2)^2=182/12-147/12=35/12. This is distinct from E[X²].
Worked example 2

Two quantities moving against each other

X and Y have variances 4 and 9, and covariance −1. What is the variance of X+Y?

  1. Keep the covariance term

    The problem supplies dependence information, so do not simply add 4 and 9.

  2. Substitute into the sum rule

    Var(X+Y)=4+9+2(−1)=11.

  3. Check the direction

    Negative covariance reduces the total spread relative to the zero-covariance value 13. It does not make the variance negative.

Concept checks

Choose an answer, then check it. You can retry; your latest checked answer determines completion.

0 of 3 concept checks passed
Concept check 1Var(X)=5. What is Var(X+100)?
Concept check 2Var(X)=2. What is Var(−3X)?
Concept check 3Which statement is valid for variables with finite second moments?

Key results

  • Average squares before subtracting the squared mean.
  • Constant shifts do not affect variance.
  • Scale factors are squared.
  • Look for a covariance term before adding spreads.

Practice problems

Solve related problems or try a timed probability drill.

Practice problemsHints and worked solutions included.Timed probability drillShort questions on this module’s topic.
Next in the course

Reason about the wait

Use geometric trials, memorylessness, and small state models.

Next module