Variables can be related in various ways. Some of these can be described mathematically. Often, a scatter plot of two variables can help to illustrate the type of relationship between them. There are also statistical tools for testing various relationships.
Negative Versus Positive Relationships
Some pairs of variables are related positively. This means that as one variable goes up, the other tends to go up as well. For example, height and weight are positively related because taller people tend to be heavier. Other pairs are negatively related, which means that as one goes down the other tends to go up. For example, gas mileage and the weight of a car are negatively related, because heavier cars tend to get lower mileage.
Linear and Nonlinear Relationships
Two variables may be related linearly. This means that a straight line can represent their relationship. For example, the amount of paint needed to paint a wall is linearly related to the area of the wall. Other relationships cannot be represented by a straight line. These are called nonlinear. For example, the relationship between height and weight in humans is nonlinear, because doubling height usually more than doubles weight. For example, a child may be three feet tall and weigh 50 pounds, but probably no six-foot tall adult weighs only 100 pounds.
Monontonic and Nonmonotonic Relationships
Relationships can be monotonic or non-monotonic. A monotonic relationship is one where the relationship is either positive or negative at all levels of the variables. A non-monotonic relationship is one where this is not so. All of the examples above were monotonic. An example of a non-monotonic relationship is that between stress and performance. People with a moderate amount of stress perform better than those with very little stress or those that have a great deal of stress.
Strong and Weak Relationships
A relationship between two variables may be strong or weak. If the relationship is strong, it means that a relatively simple mathematical formula for the relationship fits the data very well. If the relationship is weak, then this is not so. For example, the relationship between the amount of paint and the size of wall is very strong. The relationship between height and weight is weaker.