Scatter Plots

A scatter plot places each paired numerical observation $(x,y)$ as one point to study how two variables vary together.

Paired data

Each point must join measurements from the same individual, object, place, or time. A student’s study time and test score form a pair; sorting each column separately would destroy those relationships.

Choose the explanatory variable for the horizontal axis and response variable for the vertical axis when that distinction makes sense. Label both axes with units and use scales that show the data honestly.

Direction

A positive association means larger $x$-values tend to accompany larger $y$-values. A negative association means larger $x$ tends to accompany smaller $y$. No association means no clear directional pattern.

Direction describes a tendency, not a rule that every point must follow.

Form and strength

Form may be linear, curved, clustered, or more complicated. Strength describes how tightly points follow that form. A strong curved association can have weak linear correlation if the wrong summary is used.

Describe a plot with direction, form, strength, and unusual features rather than saying only “there is correlation.”

Outliers and influential points

An outlier lies away from the main pattern. Check for data-entry error, unusual context, or a legitimate exceptional case. An influential point is one whose removal substantially changes a fitted model; it may be far in the $x$-direction even if it follows the trend.

Do not delete points merely to improve a line. Document and justify any exclusion.

Lines of best fit

For an approximately linear pattern, a line of best fit summarizes predicted response:

Slope estimates change in predicted $y$ per unit of $x$. The intercept is the predicted response at $x=0$, but it may have no practical meaning if zero is outside the observed domain.

Residuals

A residual is observed minus predicted:

Positive residuals lie above the fitted line and negative residuals below it. Random residual scatter supports a linear model; a curved residual pattern suggests a missing nonlinear structure.

Interpolation and extrapolation

Interpolation predicts within the observed $x$-range and is usually safer. Extrapolation predicts beyond it, where the pattern may change. A model fitted to adolescent heights should not be extended indefinitely to older ages.

Report predictions with reasonable precision and the model’s domain.

Correlation is not causation

Association alone does not prove that changing $x$ changes $y$. A lurking variable may affect both, the causal direction may be reversed, or the pattern may be coincidental. Randomized experiments support causal conclusions more strongly than observational scatter plots.

Common mistakes

Connecting points in arbitrary order. A scatter plot is not automatically a time-series line graph.

Claiming causation from association. Consider study design and other variables.

Ignoring axis scale. Truncated or uneven scales can exaggerate patterns.

Extrapolating far beyond the data. The fitted relationship may not continue.

Quick self-check

• Are observations correctly paired?
• What are direction, form, strength, and unusual features?
• Is a linear model appropriate?
• Is my prediction interpolation or extrapolation?
• Am I distinguishing association from causation?

Related topics

• Correlation
• Linear Regression
• Types of Data
• Sampling