scatterplots

Scatterplots are a two-dimensional graph used to visualize the relationship between two numerical variables. Each point represents an observation, with its x-coordinate given by the value of the first variable and its y-coordinate by the value of the second. They are a standard tool in exploratory data analysis.

Scatterplots help identify the direction (positive, negative), the form (linear, curved, or more complex), and the strength of association. They can reveal clusters, patterns, and outliers, as well as heteroscedasticity, where the spread of points changes with the level of the variables.

Interpreting a scatterplot often involves fitting a summary line, such as a least-squares regression line or a locally estimated scatterplot smoothing (LOESS) curve. The strength of a linear relationship is commonly quantified by the correlation coefficient r, and the fraction of variance explained is r^2. Causation cannot be inferred from correlation alone.

Variations include scatter plot matrices that display pairwise relationships among many variables, and three-dimensional scatter plots that add a z-coordinate. Visual encodings such as color, size, or symbol shape can represent a third variable, increasing information content.

Scatterplots are widely used to guide model selection, detect anomalies, and communicate relationships in disciplines ranging from science to economics. They are simple to construct but can be sensitive to outliers and sample size, and care is needed in interpretation.