Relationship between two numerical variables

Scatterplots, correlation coefficients and least squares lines honestly summarise the relationship between two numerical variables, Y and X.

However they can sometimes give a misleading impression about the relationship. The problem arises when other variables are also associated with both Y and X.

Marginal relationship between Y and X
The relationship that is evident in a scatterplot of the two variables without taking into account any other variables.
Conditional relationship between Y and X, given Z
The relationship between Y and X that is evident within subsets of the data corresponding to different values of Z.

The marginal and conditional relationships between Y and X are often different and may even be in a different direction.

Lurking (or hidden) variables

If the marginal relationship between X and Y is different from their conditional relationship given Z, but Z has either not been recorded or is ignored when analysing the data, then Z is called a lurking variable (or a hidden variable).

Always think about whether there might be a lurking variable, Z, that is distorting the relationship that is observed between Y and X.