Representing the Relationship Between Two Quantitative Variables - AP Statistics

SECTION 1

Historical Context & Motivation

The desire to understand how two measurements relate to each other predates modern statistics by centuries. Early astronomers plotted celestial observations against time, hoping to detect predictable patterns in planetary motion, but it was not until the nineteenth century that formal graphical and algebraic tools emerged for quantifying bivariate relationships. The scatterplot—a graph of paired observations on a Cartesian plane—became the primary visual device for displaying how one quantitative variable changes with another. Alongside it, the concept of correlation gave statisticians a single number to summarize the strength and direction of a linear association. These innovations transformed data analysis from purely narrative descriptions into a precise, reproducible science.

1686 • First Known Scatterplot

Edmund Halley plotted barometric pressure against altitude, producing one of the earliest documented scatterplots and demonstrating that graphical displays of paired data could reveal physical laws.

1801 • Playfair's Statistical Graphics

William Playfair published time-series and bar charts in his Commercial and Political Atlas, popularizing the idea that numerical relationships are best communicated visually rather than in tabular form.

1885 • Galton and Regression

Francis Galton coined 'regression toward mediocrity' while studying hereditary stature, creating scatterplots of parent and offspring heights and fitting median lines to the data.

1896 • Pearson's Correlation Coefficient

Karl Pearson formalized the product-moment correlation coefficient r, providing a standardized measure of the linear association between two quantitative variables that remains the most widely used today.

The central question these pioneers tried to answer remains the same one you face in AP Statistics: given a set of paired numerical observations, how can we describe and quantify the pattern—or lack thereof—between the two variables? Addressing this question requires both graphical tools (scatterplots) and numerical summaries (direction, form, strength, and unusual features), which together lay the groundwork for the least-squares regression models you will study next.

SECTION 2

Core Principles & Definitions

Before diving into computations, it is essential to establish the vocabulary and conceptual framework that AP Statistics uses to describe bivariate quantitative data. Every scatterplot is built from ordered pairs (x, y), where x is the explanatory (independent) variable and y is the response (dependent) variable. The distinction matters: the explanatory variable is the one we suspect might influence or predict the response, even though a scatterplot alone never proves causation. When describing what you see in a scatterplot, AP readers expect you to comment systematically on four features.

Direction

A positive association means that as x increases, y tends to increase. A negative association means y tends to decrease as x increases. If neither trend is apparent, the variables show no association.

Form

The overall shape of the cloud of points may be linear (points cluster around a straight line), curved (e.g., quadratic or exponential), or show no clear form at all.

Strength

Strength describes how tightly the data points cluster around the underlying pattern. A strong association shows little scatter; a weak association shows considerable scatter. The correlation coefficient r quantifies strength for linear patterns.

Unusual Features

Always note outliers (points far from the overall pattern), clusters (distinct subgroups within the data), or other deviations that would influence a summary statistic.

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KEY TAKEAWAY

Think of a scatterplot as a satellite photograph of a landscape: the direction tells you whether the terrain climbs or descends, the form tells you whether the path is a straight road or a winding river, the strength tells you how closely travelers stick to the path, and unusual features are the boulders or detours that break the pattern. On the AP exam, you must address all four to earn full credit when describing an association.

SECTION 3

Anatomy of a Scatterplot

A well-constructed scatterplot is the indispensable first step in exploring bivariate quantitative data. The diagram below illustrates how paired data on study hours (explanatory variable) and exam score (response variable) are plotted, along with annotations for every feature AP readers expect you to describe.

The cyan dots represent individual students' paired observations (study hours, exam score). The dashed violet line indicates the overall linear trend. The red dot labeled Outlier falls far above the trend—this student scored much higher than expected given relatively few study hours.

Notice that the explanatory variable (study hours) is placed on the horizontal axis and the response variable (exam score) on the vertical axis. The cloud of points rises from left to right, indicating a positive direction. The points cluster tightly around a straight dashed line, suggesting a strong linear form. The single red point falls well above the overall pattern—a clear outlier that deserves mention in any description of the relationship. When writing about a scatterplot on the AP exam, always address direction, form, strength, and unusual features in context; vague language like 'the variables are correlated' without specifying these characteristics will cost points.

SECTION 4

Mathematical Framework: The Correlation Coefficient

While scatterplots provide a visual summary, the Pearson product-moment correlation coefficient r offers a numerical measure of the direction and strength of a linear relationship between two quantitative variables. Understanding its derivation from standardized values (z-scores) reveals why r has the properties it does and why it is limited to detecting linear patterns.

Z-SCORE STANDARDIZATION

zₓ = (xᵢ − x̄) / sₓ z_y = (yᵢ − ȳ) / s_y

Each observation is expressed in units of standard deviations from the mean. Here x̄ and ȳ are the sample means, and sₓ and $s_y$ are the sample standard deviations of x and y, respectively.

CORRELATION COEFFICIENT

r = (1 / (n − 1)) × Σ[((xᵢ − x̄)/sₓ) × ((yᵢ − ȳ)/s_y)]

Equivalently, r = (1/(n − 1)) × Σ(zₓ × $z_y$ ). The correlation r averages the products of the paired z-scores. When both z-scores have the same sign (both above or both below their respective means), the product is positive, pulling r toward +1. When they have opposite signs, the product is negative, pulling r toward −1.

Several essential properties follow directly from this formula. First, r is unitless because the standardization removes the original units. Second, r is bounded: −1 ≤ r ≤ 1, where r = 1 indicates a perfect positive linear relationship and r = −1 indicates a perfect negative linear relationship. Third, r is symmetric in x and y, meaning the correlation of x with y is identical to the correlation of y with x. Fourth, and crucially for the AP exam, r measures only linear association: a strong curved relationship can have r ≈ 0 if the positive and negative z-score products cancel out.

COEFFICIENT OF DETERMINATION

r² = (fraction of variability in y explained by the linear model with x)

For example, if r = 0.90, then r² = 0.81, meaning approximately 81% of the variation in y is accounted for by the linear relationship with x. The remaining 19% is due to other factors and random scatter.

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Calculator Tip

On the TI-83/84, enter x-values in L1 and y-values in L2, then run LinReg(a+bx) L1, L2. The output includes r and r². Make sure DiagnosticOn is enabled (found in the CATALOG menu) or r will not appear.

SECTION 5

Classifying Association Patterns

Not all scatterplots look the same. Recognizing common patterns—and knowing which summary statistics are appropriate for each—is a skill tested repeatedly on the AP exam. The diagram below presents six archetypal scatterplot patterns along with their approximate correlation coefficients, helping you calibrate your visual intuition.

Six archetypal scatterplot patterns. The top row shows strong positive linear, moderate negative linear, and no linear association. The bottom row shows a curved pattern where r ≈ 0 despite a strong relationship, clustered data suggesting subgroups, and an influential outlier that distorts r.

The bottom-left panel (Curved) is an especially important case for AP Statistics: the relationship between x and y is clearly strong, yet r is nearly zero because the positive and negative z-score products cancel. This underscores why you must always examine the scatterplot before relying on r. Similarly, the bottom-right panel shows how a single influential outlier can either inflate or deflate r dramatically, depending on its position. On free-response questions, noting such features and explaining their effect on r demonstrates the kind of statistical reasoning that earns full credit.

SECTION 6

Worked Example: Computing and Interpreting r

A researcher collects data on the age (in years) and resting heart rate (in beats per minute) of five adults. The data are: (25, 72), (30, 70), (35, 74), (40, 78), (50, 82). Describe the association and compute r.

Computing the Correlation Coefficient

1
Step 1 — Compute Means

Sum the x-values: 25 + 30 + 35 + 40 + 50 = 180. Mean x̄ = 180/5 = 36. Sum the y-values: 72 + 70 + 74 + 78 + 82 = 376. Mean ȳ = 376/5 = 75.2.

x̄ = 36, ȳ = 75.2
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Step 2 — Compute Standard Deviations

Deviations for x: (−11, −6, −1, 4, 14). Squared deviations sum: 121 + 36 + 1 + 16 + 196 = 370. So sₓ = √(370/4) = √92.5 ≈ 9.618. Deviations for y: (−3.2, −5.2, −1.2, 2.8, 6.8). Squared deviations sum: 10.24 + 27.04 + 1.44 + 7.84 + 46.24 = 92.80. So $s_y$ = √(92.80/4) = √23.2 ≈ 4.817.

sₓ ≈ 9.618, $s_y$ ≈ 4.817
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Step 3 — Compute z-Score Products

For each pair, compute zₓ × $z_y$ . Observation 1: (−11/9.618)(−3.2/4.817) = (−1.144)(−0.664) = 0.760. Observation 2: (−6/9.618)(−5.2/4.817) = (−0.624)(−1.079) = 0.673. Observation 3: (−1/9.618)(−1.2/4.817) = (−0.104)(−0.249) = 0.026. Observation 4: (4/9.618)(2.8/4.817) = (0.416)(0.581) = 0.242. Observation 5: (14/9.618)(6.8/4.817) = (1.456)(1.412) = 2.056.

Products: 0.760, 0.673, 0.026, 0.242, 2.056
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Step 4 — Compute r

Sum the products: 0.760 + 0.673 + 0.026 + 0.242 + 2.056 = 3.757. Divide by n − 1 = 4: r = 3.757/4 ≈ 0.939.

r ≈ 0.939
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Step 5 — Interpret in Context

There is a strong, positive, linear association between age and resting heart rate for these five adults. As age increases, resting heart rate tends to increase. The correlation r ≈ 0.939 indicates that the data points cluster tightly around a line with a positive slope. The coefficient of determination r² ≈ 0.882 means approximately 88.2% of the variation in resting heart rate is explained by the linear relationship with age.

r² ≈ 0.882 → about 88.2% of variation in heart rate is explained by the linear model.

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AP Exam Interpretation Template

When interpreting r in context on the AP exam, use this structure: 'There is a [strength], [direction], [form] association between [x in context] and [y in context].' For r²: 'Approximately [value]% of the variability in [y in context] is explained by the linear relationship with [x in context].' Never say r or r² 'causes' anything.

SECTION 7

Strengths, Limitations & Common Pitfalls

Scatterplots and the correlation coefficient are powerful tools, but each has inherent limitations that AP Statistics frequently tests. The table below organizes the key strengths alongside the most common pitfalls that lead students to misinterpret data.

Feature	Strengths	Limitations / Pitfalls
Scatterplot	Reveals direction, form, strength, outliers, and clusters at a glance. Requires no assumptions about the data.	Overplotting can obscure patterns when n is very large. Scale choices can distort perceived strength.
r (correlation)	Unitless, bounded (−1 to 1), symmetric in x and y. Summarizes linear strength and direction in a single number.	Only captures linear association. Sensitive to outliers. Cannot distinguish cause from association.
r² (determination)	Directly interpretable as the proportion of variability in y explained by the linear model.	Does not reveal direction (always positive). A high r² for a curved relationship is misleading.
Correlation ≠ Causation	Identifying strong association motivates further investigation (experiments, controls).	Lurking variables and confounders can create spurious correlations. Only randomized experiments establish causation.

Summary of strengths and limitations for the primary tools of bivariate quantitative analysis.

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KEY TAKEAWAY

Think of r as a thermometer that only reads Fahrenheit: it tells you the 'temperature' of the linear component of an association, but it is blind to non-linear patterns the way a thermometer is blind to humidity. A scatterplot is the full weather report—always look at the graph before trusting the number. On the AP exam, the most common error is citing a high r as proof that a linear model is appropriate without first confirming linearity in the scatterplot.

SECTION 8

Connection to Regression & Advanced Models

Describing an association with a scatterplot and r is the first act in a larger statistical narrative. The natural next step is to model the relationship with the least-squares regression line (LSRL), which produces a prediction equation ŷ = a + bx. The slope b is directly linked to r through the formula b = r × ( $s_y$ / sₓ), revealing that the correlation determines the rate of change in predicted y per unit change in x after adjusting for the spread of both variables. Beyond AP Statistics, the framework generalizes to multiple regression (several explanatory variables), logistic regression (binary response), and non-parametric smoothers that relax the linearity assumption entirely.

Concept	In This Lesson (Descriptive)	Next Topic (Modeling)
Graphical tool	Scatterplot (displays raw data)	Residual plot (displays model fit)
Numerical summary	r (correlation), r² (determination)	a (intercept), b (slope), s (standard deviation of residuals)
Goal	Describe the association	Make predictions and quantify uncertainty
Assumptions	None (purely descriptive)	Linearity, independence, normal residuals, equal variance (LINE)

How descriptive bivariate analysis connects to the LSRL and inference topics.

Mastering the material in this lesson—scatterplot construction, DFSOU (direction, form, strength, outliers/unusual features) descriptions, and the properties of r—provides the conceptual scaffolding for every regression topic that follows. In particular, recognizing that r only measures linear association will help you decide when a linear model is appropriate and when transformations (logarithmic, power) should be considered.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

A scatterplot of the number of hours of television watched per day (x) and grade point average (y) for a sample of college students shows a moderately strong, negative, linear association with no outliers. Which of the following is the best interpretation of this association?

A. Watching more television causes students to have lower GPAs.
B. Students who watch more hours of television tend to have lower GPAs, but this does not establish a causal relationship.
C. There is no meaningful relationship between television watching and GPA because correlation does not imply causation.
D. GPA causes students to change how much television they watch.
E. The association must be weak because observational data cannot produce strong correlations.

Show answer (B)

The correct answer is B. A negative association means that higher values of x (TV hours) correspond to lower values of y (GPA), and 'moderately strong' means the points cluster somewhat tightly around the trend. However, because this is observational data, we can describe the tendency but cannot conclude causation. Choice A is wrong because it makes a causal claim from an association. Choice C incorrectly dismisses the relationship entirely—correlation not implying causation does not mean there is no meaningful association, only that causation is not established. Choice D reverses the explanatory/response roles and also makes a causal claim. Choice E is false because observational data can absolutely produce strong correlations.

Question 2: BASIC CALCULATION

For a bivariate data set, r = −0.80. What is the value of r², and what does it tell us about the relationship?

A. r² = 0.64; about 64% of the variability in y is explained by the linear relationship with x.
B. r² = −0.64; about 64% of the variability in y is explained by the negative linear relationship with x.
C. r² = 0.80; about 80% of the variability in y is explained by the linear relationship with x.
D. r² = 0.64; about 64% of the data points fall exactly on the regression line.
E. r² = 0.40; about 40% of the variability in y is explained by the linear relationship with x.

Show answer (A)

The correct answer is A. r² = (−0.80)² = 0.64, meaning approximately 64% of the variability in the response variable is explained by the linear model with the explanatory variable. Choice B is incorrect because r² is always non-negative—squaring a negative number yields a positive result. Choice C confuses r with r²; it uses the value of r rather than squaring it. Choice D gives the correct numerical value of r² but misinterprets it—r² represents the proportion of variability explained, not the percentage of points on the line. Choice E incorrectly computes r² (perhaps confusing it with |r|/2 or some other operation).

Question 3: INTERMEDIATE

A researcher measures the speed (in miles per hour) and fuel efficiency (in miles per gallon) for 30 different car models. The scatterplot shows a curved pattern: fuel efficiency increases as speed increases from 15 to 45 mph, then decreases as speed increases from 45 to 80 mph. The correlation is r = −0.10. Which of the following is the best conclusion?

A. Speed and fuel efficiency have virtually no association.
B. The weak correlation proves that speed does not affect fuel efficiency.
C. There is a strong non-linear association between speed and fuel efficiency, but r fails to detect it because r only measures linear association.
D. The negative value of r confirms a negative association at all speeds.
E. The researcher should discard the data because a correlation near zero indicates data collection errors.

Show answer (C)

The correct answer is C. The scatterplot clearly reveals a strong curvilinear (quadratic-like) pattern, yet r ≈ −0.10 because the positive z-score products from the rising portion and the negative products from the falling portion largely cancel each other out. This is a classic illustration that r only captures linear association. Choice A is wrong because there IS a strong association—it's just not linear. Choice B makes a causal claim and incorrectly equates a weak r with no association at all. Choice D misinterprets r; the relationship is positive at low speeds and negative at high speeds, not uniformly negative. Choice E is absurd—a near-zero r can arise legitimately from non-linear data and is not evidence of data collection errors.

Question 4: APPLIED

4 pts

An environmental scientist collects data on the average daily temperature (°F) and the dissolved oxygen concentration (mg/L) in a lake over 20 days. She produces a scatterplot that shows a strong, negative, roughly linear association with one point that falls far from the pattern. Using her calculator, she finds r = −0.87 with all 20 observations and r = −0.95 after removing the unusual point. (a) Describe the association shown in the scatterplot in context, addressing direction, form, strength, and unusual features. (b) Explain why r changed when the unusual point was removed. (c) Should the scientist automatically remove the unusual point before computing the least-squares regression line? Justify your answer. (d) Interpret the value r² = 0.7569 (from all 20 observations) in context.

Show answer

Rubric (4 points): Point 1 (Description of association): There is a strong, negative, linear association between average daily temperature and dissolved oxygen concentration in the lake. As temperature increases, dissolved oxygen tends to decrease. There is one unusual point (outlier) that does not follow the overall pattern. Point 2 (Explanation of r change): The unusual point weakens the linear pattern by adding scatter that does not fit the overall negative trend. Because r is sensitive to outliers—particularly points with extreme x-values or y-values—removing this influential point allows the remaining data to cluster more tightly around the linear trend, pulling r from −0.87 to −0.95. Point 3 (Judgment on removing the point): The scientist should NOT automatically remove the outlier. She should first investigate whether it resulted from a recording error, instrument malfunction, or an unusual condition (e.g., a day with unusual runoff). If a legitimate reason for exclusion is found, removal is justified; otherwise, the point represents a real observation and should be retained. Removing data solely to improve r is statistically inappropriate. Point 4 (Interpretation of r²): Approximately 75.69% of the variability in dissolved oxygen concentration in the lake is explained by the linear relationship with average daily temperature. The remaining approximately 24.31% of the variability is due to other factors not included in the model.

Question 5: CRITICAL THINKING

4 pts

A statistics student computes the correlation between the number of firefighters sent to a fire (x) and the property damage in thousands of dollars (y) for a random sample of 50 fires in a city. She obtains r = 0.92 and concludes: 'Sending more firefighters causes more property damage. The fire department should send fewer firefighters to reduce damage.' (a) Explain the flaw in the student's reasoning. (b) Identify a lurking variable that could explain the strong positive correlation between number of firefighters and property damage. (c) Describe how you would design a study that could establish whether the number of firefighters has a causal effect on property damage, and explain why your design would be more convincing than the observational data. (d) Even with the student's flawed reasoning set aside, explain one additional limitation of using r to summarize this data set.

Show answer

Rubric (4 points): Point 1 (Flaw in reasoning): The student confuses correlation with causation. The data are observational, so the strong positive correlation between firefighters sent and property damage does not establish that sending more firefighters causes more damage. Association alone cannot prove a causal relationship because there may be confounding variables that drive both quantities simultaneously. Point 2 (Lurking variable): The severity (size) of the fire is a lurking variable. Larger, more intense fires both require more firefighters to be dispatched and inherently cause more property damage. The fire's severity confounds the relationship—it is the common cause behind both x and y, producing a positive correlation even though firefighters are working to reduce damage. Point 3 (Study design): An experiment would be needed, but randomly assigning different numbers of firefighters to fires of similar severity raises serious ethical and practical concerns. A feasible alternative would be a matched observational study: compare fires of similar initial severity (matched on response time, building type, and initial flame size) but where varying numbers of firefighters happened to be available. While this does not fully establish causation like a randomized experiment, it controls for the primary confounder (fire severity) and would provide much stronger evidence than the raw correlation because it reduces confounding. Point 4 (Additional limitation of r): The relationship between firefighters sent and property damage may not be linear. For instance, very small fires may show minimal damage regardless of crew size, while very large fires may show a plateau in damage. If the scatterplot reveals a non-linear pattern, r would understate the true strength of the association, and the student's reliance on a single linear summary statistic would miss important features of the data.

SUMMARY

Lesson Summary

Summary

To represent the relationship between two quantitative variables, begin by constructing a scatterplot with the explanatory variable on the horizontal axis and the response variable on the vertical axis. When describing the pattern, address all four characteristics: direction (positive, negative, or neither), form (linear, curved, or no pattern), strength (how tightly points cluster around the pattern), and unusual features (outliers, clusters, influential points). Always describe these characteristics in the context of the variables.

The correlation coefficient r quantifies the direction and strength of a linear association, ranging from −1 to +1. Its square, r², gives the proportion of variability in y explained by the linear model. Remember that r is sensitive to outliers, blind to non-linear patterns, and does not imply causation. Always examine the scatterplot before computing or interpreting r, and always describe associations in the context of the data.