A:

The correlation coefficient can be used to measure the linear dependence between two random variables. The most common correlation coefficient, generated by the Pearson product-moment correlation, may be used to measure the linear relationship between two variables. However, in a non-linear relationship, this correlation coefficient may not always be a suitable measure of dependence.

The differences between correlation and dependence may be illustrated by the concepts of correlation and causation. The coefficient of correlation does not indicate the presence of a causal relationship between two variables. For example, there is no proven causal relationship between happiness and physical strength. While an analysis of data may indicate a positive correlation between the two variables, it does not imply that happiness causes an increase in physical strength or its converse — that an increase in physical strength causes happiness — are true. Consequently, the dependence of one variable on the other cannot be ascertained directly from the coefficient of correlation because of the action of extraneous random variables that influence statistical dependence. For example, the correlation between the number of sailors on a ship and its average speed does not indicate causation due to the presence of several other factors, such as the weather conditions, throttle settings and its payload. The financial industry also uses the principles of causation and correlation in relation to the relationship between earnings per share (EPS) and other financial metrics.

There are several types of correlation coefficients used to determine the relationship between varied data types, including the Spearman rank-order correlation, Biserial correlation and Phi correlation. The Pearson coefficient of correlation is denoted by the letter "r" and may be used to interpret the strength or weakness of a relationship between two variables between the values +1 and -1. When squared, the resulting value is known as the coefficient of determination that expresses the variation of such a relationship.

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3. ### When does positive correlation prove causation?

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