4. Exploring Relationships in Data

Match the Analysis to the Variable Types

The first step in analyzing a relationship is to classify the variables. A quantitative variable records numerical measurements for which differences have meaningful interpretations, such as age, income, temperature, or exam score. A categorical variable records labels or groups, such as treatment group, region, or transportation type.

Choose the display and summary that match the variable pairing:

• For one quantitative variable and one categorical variable, compare group distributions with side-by-side boxplots, grouped dotplots, or grouped histograms. Means or medians can summarize the groups.

• For two quantitative variables, use a scatterplot, correlation, or simple linear regression.

• For two categorical variables, use a two-way table, conditional percentages, or a clustered or stacked bar chart.

Category codes are labels, not automatically meaningful numerical measurements. Therefore, do not calculate a correlation between arbitrary category codes and a quantitative outcome.

Takeaway: Identify the variable types before selecting a graph, numerical summary, or model.

Compare Groups with Quantitative and Categorical Variables

When one variable is quantitative and the other is categorical, compare the distribution of the quantitative variable across the groups. For example, hours of sleep is quantitative, while class standing is categorical. The relevant question is how sleep differs among first-year students, sophomores, juniors, and seniors—not whether numerical codes assigned to class standing correlate with sleep.

Side-by-side boxplots allow comparison of:

• center, such as the median;

• spread, such as the interquartile range;

• shape and possible skewness; and

• unusual observations or outliers.

A responsible conclusion might be: “In this sample, first-year students tended to report more hours of sleep than seniors, although the distributions overlapped.” This describes an observed group difference without claiming that progressing through college causes students to sleep less.

A categorical variable can also identify groups within a scatterplot. Different colors or symbols might distinguish online and in-person sections in a plot of study hours and exam scores. Examine both the combined pattern and the within-group patterns, because differences between groups can conceal, weaken, or reverse a relationship.

Read Relationships from Scatterplots

A Scatterplot displays paired quantitative observations as points $(x,y)$. Each point represents one case, such as a student, household, day, or business.

Construct the graph carefully:

1. Put the explanatory variable on the horizontal axis.

2. Put the response variable on the vertical axis.

3. Label both axes and include units.

4. Choose a scale that makes the pattern visible without exaggerating it.

Describe the graph using four features:

• Direction: A positive association means larger values of $x$ tend to occur with larger values of $y$; a negative association means larger values of $x$ tend to occur with smaller values of $y$.

• Form: Decide whether the pattern is approximately linear, curved, clustered, or otherwise structured.

• Strength: Assess how closely the points follow the general pattern.

• Unusual observations: Look for outliers or influential points.

For example, advertising expenditure and sales might show a moderately strong, positive, roughly linear association. That description does not imply that every business follows the same pattern or that advertising alone caused higher sales.

Takeaway: Inspect the graph before calculating a numerical summary; curvature, clusters, and influential points can be hidden by a single number.

Interpret Correlation and Explained Variation

Correlation measures the direction and strength of a linear association between two quantitative variables. The sample correlation coefficient is written $r$, and it satisfies

Interpret its value as follows:

• $r>0$: positive linear association;

• $r<0$: negative linear association;

• $r\approx 0$: little or no linear association;

• $|r|$ near $1$: strong linear association; and

• $|r|$ near $0$: weak linear association.

For example, $r=0.82$ indicates a strong positive linear association, whereas $r=-0.25$ indicates a weak negative linear association.

Correlation has important limits:

• It describes linear association only. A strong curved relationship can have a correlation near zero.

• It is not appropriate when one variable is merely a set of category labels.

• It can be strongly affected by outliers.

• It does not identify which variable influences the other.

• It does not establish causation.

The coefficient of determination is $r^2$. If $r=0.80$, then

The appropriate interpretation is that about $64\%$ of the variation in the response is accounted for by its linear association with the explanatory variable in the sample. The remaining variation may reflect other variables, random variation, measurement error, or departures from the linear model.

Use Linear Regression to Predict

Simple linear regression predicts a quantitative response from one quantitative explanatory variable. The fitted regression equation is

where $\hat{y}$ is the predicted response, $b_0$ is the fitted intercept, $b_1$ is the fitted slope, and $x$ is the explanatory-variable value.

The slope describes the predicted average change in $y$ for a one-unit increase in $x$. For example, in

the model predicts an average increase of $4.2$ exam-score points for each additional hour studied, within the range of study hours represented in the data.

The intercept is the predicted response when $x=0$. Interpret it only when zero is meaningful and is within, or reasonably near, the observed range. If the observed study times range from $2$ to $12$ hours, interpreting the intercept as a realistic prediction for zero hours may be inappropriate.

The usual fitted line is the least-squares regression line. It minimizes the sum of squared vertical differences between observed and predicted responses. These differences are Residual values:

A positive residual places the observation above the line, a negative residual places it below the line, and a zero residual means the prediction is exact. A residual plot should show points scattered randomly around zero. Curvature suggests that a straight-line model is inadequate; a funnel shape suggests that variability changes as the explanatory variable changes.

Recognize the Risks of Extrapolation

A regression equation is generally most trustworthy for explanatory-variable values within the range used to fit the model. Extrapolation occurs when a model is used outside that range.

For example, a model based on temperatures from $60^\circ\mathrm{F}$ to $90^\circ\mathrm{F}$ may reasonably predict ice-cream sales at $75^\circ\mathrm{F}$. It should not automatically be used to predict sales at $10^\circ\mathrm{F}$ or $130^\circ\mathrm{F}$, because the relationship may change outside the observed temperatures.

A strong correlation or statistically significant regression slope does not make an extrapolated prediction safe. Check the range, the graph, the residuals, and the practical context before using a model for prediction.

Takeaway: A model can describe the observed range well and still give unreliable predictions beyond that range.

Separate Association from Causation

An association means that two variables vary together in a systematic way. Causation means that changing one variable produces a change in the other, with other relevant conditions held constant. Association alone does not establish causation.

An observed relationship may result from:

1. direct causation;

2. reverse causation, in which the response influences the explanatory variable;

3. Confounding, in which a third variable affects both variables;

4. coincidence; or

5. sampling variation.

For example, people who carry larger umbrellas may be more likely to wear raincoats. Carrying an umbrella does not cause someone to wear a raincoat; rainy weather is a lurking variable that influences both behaviors.

Similarly, students who study more may earn higher scores, but the association could also reflect prior preparation, access to tutoring, motivation, course difficulty, or available time. An observational study can establish an association and provide evidence consistent with causation, but it generally cannot eliminate every alternative explanation. A well-designed randomized experiment provides stronger evidence for a causal claim because treatments are assigned randomly and systematic group differences are reduced.

Match the conclusion to the design:

• In an observational study, say that the variables were associated or that one group had a higher average.

• In a randomized experiment, a causal statement may be justified when the design and analysis support it.

• For a regression model, say that the model predicts rather than claiming that the explanatory variable causes the response.

Takeaway: Neither correlation nor regression alone proves causation; the study design and possible confounding determine how strong a conclusion is justified.

Apply a Complete Relationship-Analysis Strategy

Use this sequence when analyzing relationships in a dataset:

1. Classify each variable as quantitative or categorical.

2. Clarify whether the goal is group comparison, description of association, or prediction.

3. Select a suitable display, such as grouped boxplots for one quantitative and one categorical variable or a scatterplot for two quantitative variables.

4. Describe the visual pattern, including direction, form, strength, clusters, group differences, and unusual observations.

5. Calculate appropriate summaries, such as group means or medians, correlation, or a regression equation.

6. Check conditions by looking for nonlinearity, unequal spread, outliers, dependence, and a restricted range.

7. Interpret the result in context, including units and the population or sample under discussion.

8. Consider the study design and possible confounding before making a causal statement.

9. Avoid overreach, including unsafe extrapolation and treating statistical detectability as practical importance.

For example, suppose a sample of homes produces a positive, roughly linear relationship between house size and annual energy use, with $r=0.76$ and the fitted model

A suitable interpretation is: In this sample, larger homes tended to use more energy, and the association was moderately strong and positive. The model predicts an additional $8.5$ units of annual energy use for each additional $100$ square feet of house size, on average. Because the data are observational, this association alone does not show that increasing house size causes increased energy use; insulation, climate, occupancy, and heating system may also matter.

This approach connects the graph, numerical summaries, units, model limitations, and study design in one defensible conclusion.