By Peter Flom • Updated Aug 30, 2022
Variables interact in a variety of ways, and many of these interactions can be described mathematically. A scatter plot often visualizes the nature of a relationship, while statistical tests confirm its significance.
When one variable increases, the other also tends to increase, we have a positive relationship. Height and weight illustrate this: taller individuals usually weigh more. In contrast, a negative relationship means that as one variable rises, the other falls. Gas mileage versus vehicle weight is a classic example—heavier cars tend to have lower fuel efficiency.
Linear relationships can be captured by a straight line. The amount of paint required to cover a wall is linearly related to the wall’s surface area; doubling the area doubles the paint needed.
Non‑linear relationships cannot be expressed by a straight line. Human height and weight exhibit a non‑linear pattern: doubling height more than doubles weight, so a six‑foot adult rarely weighs only 100 pounds.
A monotonic relationship maintains the same direction—always positive or always negative—across all levels of the variables. The examples above are monotonic. A non‑monotonic relationship, however, changes direction; for instance, performance often peaks at moderate stress levels and declines when stress is either too low or too high.
The strength of a relationship reflects how well a simple mathematical model fits the data. The paint‑to‑wall‑area correlation is strong—most of the variability in paint volume is explained by wall size. In contrast, the height‑weight correlation is weaker; many other factors influence body weight.
Statistical tools—such as correlation coefficients, regression analysis, and hypothesis testing—quantify these relationships and help researchers assess their reliability.
