Relationships between two variables
Many questions ask how two variables relate: does studying more raise scores? Scatterplots and correlation describe such relationships.
Scatterplots
A scatterplot shows two quantitative variables, one on each axis, with a point for each individual. Put the explanatory variable (the one that may influence) on the \(x\)-axis and the response variable on the \(y\)-axis. Look for direction (positive or negative), form (linear or curved), strength (how tightly points cluster), and outliers.
The correlation coefficient
Correlation \(r\) measures the strength and direction of a linear relationship, from \(-1\) to \(1\). Values near \(\pm 1\) mean points lie close to a straight line; values near \(0\) mean no linear pattern. \(r\) has no units and does not change if you rescale either variable, for example from centimetres to inches.
-1 \le r \le 1Reading correlation values
Rough guidelines: \(|r| > 0.8\) is strong, \(0.5\) to \(0.8\) moderate, below \(0.3\) weak. But context matters, and \(r\) only measures linear association. A perfect U-shaped relationship can have \(r = 0\). Always plot the data before computing \(r\).
Correlation is not causation
Ice cream sales and drowning deaths are positively correlated, but ice cream does not cause drowning; hot weather drives both. A lurking (confounding) variable can create a correlation without causation. Only a well-designed randomised experiment can establish cause and effect convincingly.
Outliers and influential points
A single unusual point can change \(r\) dramatically. A point far to the right or left of the others can pull the trend strongly; such points are called influential. Check whether conclusions change when an unusual point is removed, and investigate why it is unusual.
Two-way tables for categorical data
For two categorical variables, a two-way table counts individuals in each combination, such as smoker/non-smoker by disease/no disease. Compare conditional percentages: what percent of smokers have the disease versus non-smokers. Simpson's paradox warns that a pattern within every subgroup can reverse when the groups are combined.
Worked example
A study of 10 students finds \(r = 0.85\) between hours studied and exam score. This indicates a strong positive linear association. It does not prove studying causes higher scores, since motivated students may both study more and do better for other reasons, but it is consistent with that explanation.
Common mistakes
Computing \(r\) for categorical data. Interpreting \(r = 0\) as 'no relationship' when the relationship is curved. Concluding causation from correlation. Ignoring influential points.
Key takeaways
Scatterplots show direction, form, strength, and outliers. Correlation \(r\) measures linear association between \(-1\) and \(1\), is unit-free, and is sensitive to outliers. Correlation does not imply causation. Two-way tables and conditional percentages handle categorical pairs.
Flashcard
Points showing two quantitative variables
Check your understanding
Cities with more firefighters have more fire damage. The most likely explanation is: