ECMT1020 Chap.4 Variance, Covariance and Correlation
Variance, Covariance and Correlation
Variance measures squared dispersion around a mean, covariance measures signed co-movement, and correlation rescales covariance by both standard deviations. This chapter derives each measure from moments and follows constants and scale factors through linear transformations. It also distinguishes a numerical linear-association summary from a causal or independence claim, and uses units and bounds as quick arithmetic checks.
The sign of a centred product explains covariance: observations above both means or below both means contribute positively, while opposite-side observations contribute negatively. Correlation preserves that sign and removes units, but it describes only linear co-movement. For sums of random variables, the covariance cross term determines whether combined uncertainty rises or falls relative to adding separate variances.
Constants and multipliers must therefore be tracked before numerical substitution, especially when negative coefficients change the cross-term sign.
What this chapter covers
- 01
Variance from moments
- 02
Standard deviation and units
- 03
Covariance
- 04
Correlation
- 05
Linear transformations
- 06
Variance of linear combinations
Worked example · free
Combine two uncertain quantities
- +1The standard deviations are two and three.
- +1Correlation is -3 divided by 2×3, which is -0.5.
- +1Use Var(aX+bY)=a²Var(X)+b²Var(Y)+2abCov(X,Y).
- +1Substitution gives 4(4)+9+4(-3).
- +1The arithmetic is 16+9-12=13.
- +1The negative covariance reduces the combined variance.
Key terms
- Variance
- Expected squared distance from a variable's mean.
- Standard deviation
- The square root of variance in the variable's original units.
- Covariance
- Expected product of two centred variables.
- Correlation
- Covariance rescaled by the two standard deviations.
- Linear combination
- A weighted sum of variables.
- Centred product
- The product of two deviations from their respective means. Same-sign deviations contribute positively to covariance, opposite-sign deviations contribute negatively, and the distribution-weighted balance of those contributions determines the covariance sign.
- Cross term
- The covariance contribution in the variance of a sum or weighted combination. It contains both coefficients and a factor of two, and it can raise or lower total variance depending on coefficient and covariance signs.
- Scale invariance
- The correlation property that positive unit changes leave its magnitude unchanged because covariance and standard deviations rescale together, making correlation unit free.
Variance, Covariance and Correlation FAQ
Why can covariance be hard to compare?
Its size depends on the units and scales of both variables. Correlation removes those units, making linear association easier to compare across settings with different measurement scales.
Can variance be negative?
No. It averages squared deviations and must be nonnegative. A negative calculated variance signals an arithmetic or formula error that should be repaired before interpretation.
Does correlation prove causation?
No. Correlation measures linear co-movement. Confounding, reverse direction, selection and chance can all produce correlation without the proposed causal relationship being valid for the population.
Exam move
Memorise the transformation rules through reasoning rather than isolated formulas: adding a constant does not change dispersion, multiplying by a scales standard deviation by its absolute value and variance by a squared. Recalculate covariance cross terms carefully, then use the correlation bound from minus one to one as an error check. Organise revision around three ledgers.
The moment ledger records E(X), E(X squared), variance and standard deviation with their units. The relationship ledger records E(XY), covariance and correlation. The transformation ledger records what changes after adding constants or multiplying variables. Completing all three for one small joint distribution creates more useful connections than solving isolated formulas.
Draw a four-quadrant centred plot and assign the sign of each product before calculating covariance. This gives a geometric prediction for the answer and shows how positive and negative contributions can cancel. When computing the variance of aX+bY, write the two own-variance terms and the cross term on separate lines.
Substitute coefficient signs only after the structure is complete; this reduces lost factors of two and incorrect signs. Use dimensional analysis throughout. Variance of dollars is in dollars squared, covariance of dollars and hours has product units, and correlation has no units. If the units do not match the requested measure, the formula choice is wrong even when the arithmetic is tidy.
Practise interpretation with paired datasets that share the same covariance but have different scales, then observe why their correlations differ. Also compare a strong curved relationship with weak linear correlation to keep association measures within their scope. Finish each problem with two checks: variance is nonnegative and correlation lies between minus one and one.
For verbal answers, name direction, linearity and the limits of the evidence. Never let a positive correlation become a causal claim without a separate design argument.
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