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. Conaffected by outliers.

1. The covariance between two variables \(X\) and \(Y\) is defined as \[ \operatorname{Cov}(X, Y) = E\left[(X - \mu_X)(Y - \mu_Y)\right], \] where \(\mu_X\) and \(\mu_Y\) are the means of \(X\) and \(Y\), respectively. 2. In contrast, the correlation coefficient is computed by normalizing the covariance by the product of the standard deviations of the two variables: \[ \rho_{X,Y} = \frac{\operatorname{Cov}(X, Y)}{\sigma_X \sigma_Y}. \] This normalization removes the units and scales the measure so that \(\rho_{X,Y}\) always lies between \(-1\) and \(+1\). 3. Evaluating the provided statements: - **"Covariance only applies to linear relationships."** Covariance indeed measures the degree to which two variables change linearly, but it is not correct to say it only applies to linear relationships. Both covariance and correlation quantify linear association. - **"Covariance ranges from -1 to +1."** This statement is incorrect. Covariance is not bounded within a fixed range; its value depends on the units and scales of the variables. It could be any real number. On the other hand, the correlation coefficient is confined within \(-1\) and \(+1\). - **"Covariance depends on the units of measurement."** This is correct. Since covariance is calculated using the actual values of the variables, its magnitude changes with the units. For example, if one variable is measured in meters versus centimeters, the covariance will differ. - **"Affected by outliers."** Both covariance and correlation can be strongly influenced by extreme values (outliers) because they involve averaging products of deviations from means. 4. In summary, the key differences are: - **Units and Scale:** Covariance has units derived from the variables, while correlation is unitless. - **Range:** Correlation is standardized to lie between \(-1\) and \(+1\), whereas covariance does not have a fixed range. - **Interpretability:** Correlation provides a more interpretable measure of the strength and direction of a linear relationship irrespective of the scales of the variables. These points highlight the fundamental distinctions between covariance and correlation.