How can a non-linear relationship among two variables influence the correlation coefficient?

How can a non-linear relationship among two variables influence the correlation coefficient? For example, ability to pay attention and the level of stimulation in the environment do not demonstrate a linear relationship. At some point, too much stimulation will limit the ability to focus and pay attention. What would happen to the size and the sign of the correlation coefficient if we correlated these two variables? How can researchers work around this issue?

Correlation analysis is the statistical tool that is used to describe the degree to which onevariable is linearly related to another. Two variables are said to be positively core-related ifthe...