Probability & Statistics

Correlation Under Shifts and Scales

Given corr(X,Y) = rho, both corr(X+5,Y) and corr(5X,Y) equal rho, because covariance subtracts the mean and a factor of 5 scales numerator and denominator alike.

solvedeasy1 min

The correlation between XX and YY is ρ\rho. What is the correlation between X+5X + 5 and YY? What is the correlation between 5X5X and YY?

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#Correlation ignores location

Correlation is covariance normalised by the two standard deviations,

ρ(X,Y)=Cov⁡(X,Y)sd⁡(X) sd⁡(Y).(1)\rho(X, Y) = \frac{\Cov(X, Y)}{\sd(X)\, \sd(Y)}. \tag{1}

Adding a constant moves XX bodily without changing how it varies around its mean. Covariance already subtracts that mean, so

Cov⁡(X+5,Y)=E[(X+5−E[X+5])(Y−EY)]=E[(X−EX)(Y−EY)]=Cov⁡(X,Y),(2)\Cov(X + 5, Y) = \E\big[(X + 5 - \E[X + 5])(Y - \E Y)\big] = \E\big[(X - \E X)(Y - \E Y)\big] = \Cov(X, Y), \tag{2}

and sd⁡(X+5)=sd⁡(X)\sd(X + 5) = \sd(X). Both the numerator and denominator are untouched, so

ρ(X+5,Y)=ρ.(3)\rho(X + 5, Y) = \rho. \tag{3}

#Positive scaling cancels

Multiplying by 55 scales the spread of XX, hence stretches covariance and standard deviation by the same factor,

Cov⁡(5X,Y)=5 Cov⁡(X,Y),sd⁡(5X)=5 sd⁡(X).(4)\Cov(5X, Y) = 5\,\Cov(X, Y), \qquad \sd(5X) = 5\,\sd(X). \tag{4}

The two factors of 55 divide out,

ρ(5X,Y)=5 Cov⁡(X,Y)5 sd⁡(X) sd⁡(Y)=ρ.(5)\rho(5X, Y) = \frac{5\,\Cov(X, Y)}{5\,\sd(X)\, \sd(Y)} = \rho. \tag{5}

Only the sign of the multiplier survives. A negative factor would flip ρ\rho to −ρ-\rho, but 5>05 > 0 leaves it alone.

YX
Correlation measures the tilt of this cloud, not where it sits or how wide it is. Adding 5 slides the points sideways and multiplying by 5 stretches them out, yet the trend line keeps the same slope sign, so the correlation stays exactly rho. A negative multiplier would mirror the cloud and flip the sign.

#Invariance under increasing affine maps

Both correlations equal ρ\rho. Correlation is invariant under an increasing affine change of either variable, since it measures the shape of the linear relationship, not its position or units.