How does the covariance measure differ from the correlation measure? Covariance only applies to linear relationships. Covariance ranges from -1 to +1 . Covariance depends on the units of measurement.

1. The covariance between two random variables \(X\) and \(Y\) is defined as \[ \operatorname{Cov}(X,Y)=E\big[(X-\mu_X)(Y-\mu_Y)\big], \] where \(\mu_X\) and \(\mu_Y\) are the means of \(X\) and \(Y\) respectively. It measures how much \(X\) and \(Y\) vary together. 2. The correlation between \(X\) and \(Y\) is the standardized version of covariance and is given by \[ \rho(X,Y)=\frac{\operatorname{Cov}(X,Y)}{\sigma_X\sigma_Y}, \] where \(\sigma_X\) and \(\sigma_Y\) are the standard deviations of \(X\) and \(Y\) respectively. Because of this standardization, the correlation always lies in the interval \([-1,1]\) and is unitless. 3. Comparing the measures: - **Units of Measurement:** The covariance depends on the units in which the variables are measured. If the units of \(X\) or \(Y\) change (for example, from meters to centimeters), the value of the covariance will also change. In contrast, the correlation is dimensionless and remains unaffected by such changes. - **Range:** Since covariance is not standardized, its value can lie anywhere between \(-\infty\) and \(+\infty\). On the other hand, correlation is always bounded between \(-1\) and \(+1\). - **Type of Relationship Measured:** Both covariance and correlation measure the strength and direction of a **linear** relationship between the variables. However, simply knowing the covariance does not immediately indicate the strength of the relationship unless one considers the units and scales of the variables. 4. Evaluation of the Statements: - "Covariance only applies to linear relationships." This is correct in the sense that the covariance (and hence the correlation) only captures linear dependence. They may not adequately describe nonlinear relationships. - "Covariance ranges from -1 to +1." This statement is **incorrect**; the covariance can take any value in \((-\infty,+\infty)\). It is the correlation that is bounded between \(-1\) and \(+1\). - "Covariance depends on the units of measurement." This statement is **correct**, as discussed above. In summary, the key differences are that covariance is not standardized (thus has units and an unbounded range) while correlation is a unitless measure standardized to lie between \(-1\) and \(+1\).