# Graphical Representations Between Two Quantitative Variables

> AP Statistics · AP Stats 2024-2026
> Source: https://www.owlsprep.com/study/ap-statistics-u14-graphical-representations-between-two-quantitative/

This module walks you through core graphical tools for paired quantitative data, including scatterplot interpretation, residual plot diagnostics, and segmented plots to account for confounding variables.

**Prerequisites:** [Understand univariate quantitative graphical representations (histograms, boxplots)](https://www.owlsprep.com/study/ap-statistics-u01-univariate-quantitative-graphs/); [Define basic properties of linear relationships](https://www.owlsprep.com/study/ap-statistics-u13-intro-to-linear-correlation/)

## Learning objectives

- Identify appropriate use cases for scatterplots to display bivariate quantitative data
- Interpret direction, form, strength, and outliers in a bivariate scatterplot
- Distinguish residual plots from raw data scatterplots to assess linear model fit
- Construct and interpret segmented scatterplots to control for third confounding variables

## Core Scatterplot Construction Rules

A valid scatterplot maps the explanatory (independent) variable to the horizontal x-axis, and the response (dependent) variable to the vertical y-axis. Every paired observation is plotted as a single unconnected point, with no lines joining sequential points unless explicitly tracking time for time-series data.

**Scatterplot** — A 2D visualization that displays the full distribution of bivariate quantitative observations to reveal patterns of association between two variables.

*Notation:* Paired points $(x_i, y_i)$

**Worked example:** Construct a scatterplot for the following paired data: x = hours studied for a quiz, y = quiz score out of 10, for 5 students: (1,3), (2,5), (3,7), (4,8), (5,9)

1. Label the x-axis 'Hours Studied' with a scale from 0 to 6, and y-axis 'Quiz Score' with a scale from 0 to 10.
2. Plot each paired point at its corresponding x and y coordinate, no connecting lines or extra markers.
3. Add a clear title 'Relationship Between Hours Studied and Quiz Score' to complete the plot.

**Check your understanding**

Test your understanding of scatterplot construction:

1. Which variable is plotted on the x-axis of a standard scatterplot?

   - Response variable
   - Explanatory variable
   - Categorical variable
   - Confounding variable

   *Why:* The explanatory independent variable is always mapped to the horizontal x-axis for standard bivariate analysis.

## Interpreting Scatterplot Key Features

All scatterplot interpretations must cover four standardized features: direction (positive, negative, or no association), form (linear, curved, clustered), strength (strong, moderate, weak based on how tightly points follow the form), and outliers (points that deviate drastically from the overall pattern).

- Positive association: As x increases, y tends to increase
- Negative association: As x increases, y tends to decrease
- No association: No consistent trend between x and y values

**Worked example:** Interpret the scatterplot of hours studied vs quiz score from the prior example

1. Direction: Positive association, as hours studied increases, quiz score tends to rise.
2. Form: Perfectly linear, no visible curvature or clustering.
3. Strength: Very strong, all points lie almost exactly along a straight trend line.
4. Outliers: No points deviate from the overall linear pattern.

> **Exam tip:** AP exam graders require you to explicitly reference all four features (direction, form, strength, outliers) when asked to describe a scatterplot, no partial credit for missing any of the four.

## Residual Plots for Linear Fit Assessment

A residual plot graphs residuals (calculated as $y - \hat{y}$, observed minus predicted y value) on the vertical axis against the explanatory x variable on the horizontal axis. A well-fitting linear model will show a random, scattered distribution of points with no visible pattern centered at residual = 0.

> **warning**
>
> A curved U-shape or parabolic pattern in a residual plot is definitive proof that a linear model is not appropriate for the raw data, even if the raw scatterplot looks roughly linear.

**Worked example:** Evaluate linear model fit for a residual plot that shows points forming a clear upward opening U-shape

1. First confirm the residual plot axes: x is explanatory variable, y is residual value.
2. Identify the clear curved parabolic pattern across the full range of x values.
3. Conclusion: The linear model is a poor fit, and a quadratic curved model will produce a much better fit for the data.

## Segmented Scatterplots for Confounder Control

Segmented scatterplots use different colors, shapes, or marker sizes to separate points by levels of a third categorical variable, allowing you to reveal hidden relationships that are masked when all points are plotted together. This is a common tool to eliminate the effect of confounding variables.

**Worked example:** Use a segmented scatterplot to reveal Simpson's paradox for data showing overall negative association between ice cream sales and drowning deaths, when segmented by month

1. Plot all points together first to see the spurious negative trend.
2. Segment points by season (winter, spring, summer, fall) using different marker colors.
3. Observe that within every season, the association between ice cream sales and drowning deaths is strongly positive, revealing the hidden confounding variable of outdoor temperature.

## Common pitfalls

- **Wrong:** Claiming causation directly from a scatterplot showing strong association
  - Why it fails: Scatterplots only display observational association, they cannot prove a causal relationship between variables
  - Correct: Explicitly describe the observed association, and note that causal conclusions require a controlled experimental design
- **Wrong:** Describing a residual plot using the same direction/strength language as a raw scatterplot
  - Why it fails: Residual plots are designed to eliminate the overall linear trend, so they have no inherent direction or strength to interpret
  - Correct: Only look for visible patterns (curvature, fanning, clusters) to assess if the linear model is appropriate
- **Wrong:** Ignoring influential points in the x-direction when interpreting scatterplots
  - Why it fails: A single extreme outlier far from the mean of x can completely flip the sign of the correlation coefficient and regression slope
  - Correct: Flag all x-direction outliers, and note how removing them would change the strength and direction of the association
- **Wrong:** Using a segmented scatterplot for a third quantitative variable
  - Why it fails: Segmenting by a continuous quantitative variable will produce unreadable, overcrowded marker groups
  - Correct: Reserve segmented plots for third categorical variables, and use marker size for continuous third variables if required
- **Wrong:** Adding extra lines connecting sequential points on a standard scatterplot
  - Why it fails: Connecting unrelated points creates artificial visual trends that do not exist in the underlying data
  - Correct: Only plot individual unconnected points, add a single regression trend line if explicitly required for analysis

## Cheatsheet

| Graph Type | Core Purpose | Key Diagnostic Features | Red Flag for Poor Fit |
| --- | --- | --- | --- |
| Raw Scatterplot | Display bivariate association | Direction, form, strength, outliers | No visible trend, extreme x outliers |
| Residual Plot | Assess linear model fit | Random scatter centered at 0 | Curved pattern, fanning residual spread |
| Segmented Scatterplot | Control for third categorical variable | Group-specific trends visible | Overlapping markers, no clear group separation |
| Overlaid Scatterplot | Compare two bivariate datasets | Two distinct trend groups | Groups fully overlapping with no separation |

## What's next

Mastering these graphical tools is the critical first step before calculating formal regression statistics, as you will always use plots to validate model appropriateness before running numerical calculations on the AP exam. Skipping graphical checks is one of the most common mistakes that costs students FRQ points, as College Board graders explicitly award marks for referencing residual plot patterns to justify model choices. You will now build on this foundation to learn the Pearson correlation coefficient, construct least squares regression lines, and perform formal residual analysis to quantify model fit.

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