Difficulty: Introductory-Intermediate | Prerequisites: Basic algebra, familiarity with two-variable data
Big picture: This material covers the bread and butter of describing bivariate quantitative data. You are learning how to look at a scatterplot and say something precise about it, how to use a least-squares regression line (LSRL) to make predictions, and how to measure and interpret the gap between what the model predicts and what actually happened. These skills form the backbone of inference later in the course. If you can describe associations, interpret slope in context, compute residuals, and reason about r², you are set for both the multiple-choice and free-response sections of the AP exam.
TL;DR: When two quantitative variables are plotted against each other, you describe their relationship using direction, form, strength, and context. The LSRL gives you a prediction equation; its slope tells you how much the predicted response changes per unit of the explanatory variable. Residuals measure how far off the prediction was, and r² tells you what percentage of the variation in the response is accounted for by the model.
Scatterplot
A graph that plots each observation as a point using one variable on the x-axis (explanatory) and another on the y-axis (response). Think of it as the first tool you reach for whenever you want to see whether two quantitative variables are related.
Association (direction, form, strength)
The overall pattern in a scatterplot, described along three dimensions: direction (positive or negative), form (linear, curved, or no clear pattern), and strength (how tightly the points cluster around the underlying pattern). In simple terms, this is your verbal summary of what the scatterplot looks like.
Least-squares regression line (LSRL)
The unique straight line that minimises the sum of squared residuals for a set of data. It is written as ŷ = a + bx, where a is the y-intercept and b is the slope. Think of it as the "best-fit" line through the cloud of points.
Slope (b)
The amount by which the predicted value of y changes for each one-unit increase in x. In context, this always needs to be stated with units and variable names, not just as a number.
y-intercept (a)
The predicted value of y when x = 0. Often has no practical meaning (e.g. you cannot have negative tablespoons of sugar), but it anchors the line mathematically.
Residual
The difference between the observed (actual) value and the predicted value: residual = y – ŷ. A positive residual means the actual value was higher than predicted; a negative residual means it was lower. Think of it as the prediction error for one specific data point.
r² (coefficient of determination)
The proportion of the variability in the response variable that is explained by the linear model. Stated as a percentage. If r² = 0.85, then 85% of the variation in the response is accounted for by the explanatory variable through the regression. In simple terms, it tells you how much of the scatter in y shrinks once you know x.
Every description of a scatterplot on a free-response question must hit four elements: direction, form, strength, and context (the actual variable names). Missing any one of these costs marks.
Direction: positive (upward trend) or negative (downward trend)
Form: linear, non-linear/curved, or no obvious pattern
Strength: strong, moderate, or weak, based on how tightly points follow the form
Context: refer to the actual variables (e.g. "sugar" and "freshness"), not just "x and y"
A complete example from the carnation experiment: "There is a strong, positive, linear association between the amount of sugar added (in tablespoons) and the number of hours the carnation stayed fresh."
An incomplete example: "As you add more sugar, the flowers stay fresher longer." This captures direction and context but misses form and strength.
The regression equation in the carnation study is ŷ = 180.8 + 15.8x, where x is tablespoons of sugar and ŷ is predicted hours of freshness.
The slope is 15.8. In context: "For each additional tablespoon of sugar added, the predicted number of hours a carnation stays fresh increases by 15.8 hours."
Three requirements for full credit: say "predicted" (or "on average"), give the amount of change with units, and name both variables.
An incomplete interpretation: "For each increase of 1 unit of x, the value of y goes up by 15.8." This uses x and y instead of the variable names, and it omits units.
For the flower with 2 tablespoons of sugar that stayed fresh for 204 hours:
Predicted value: ŷ = 180.8 + 15.8(2) = 180.8 + 31.6 = 212.4 hours
Residual: y – ŷ = 204 – 212.4 = –8.4 hours
Interpretation: "The carnation with 2 tablespoons of sugar stayed fresh for 8.4 fewer hours than the regression model predicted."
Key detail: the formula is always actual minus predicted (y – ŷ), never predicted minus actual. A negative residual means the model overestimated.
A common student error in the samples: one student wrote the residual as 212.4 – 204 = 8.4, which reverses the sign. The magnitude is correct, but the direction of the error is wrong, and the interpretation then says the flower lasted longer than expected when it lasted shorter.
The question: if a second group runs the same experiment but uses different varieties of flowers in addition to carnations, would r² increase, decrease, or stay the same?
Different flower varieties respond differently to sugar, introducing extra scatter around the regression line that is not explained by the sugar variable alone.
More unexplained scatter means a smaller proportion of total variability is captured by the linear model.
r² would be lower for the second group.
An incorrect response: "Because the second group is using the same number of flowers, I expect r² to stay about the same." This ignores the source of the new variability. The total sample size is the same, but the mix of flower types introduces variation that sugar alone cannot predict.
LSRL equation
ŷ = a + bx
where a = y-intercept, b = slope
Residual
residual = y – ŷ = observed – predicted
Coefficient of determination
r² = (proportion of variation in y explained by the regression on x)
The scatterplot in this study has sugar (tbs) on the horizontal axis (0 to 3) and freshness (hours) on the vertical axis (roughly 160 to 240). Points follow a clear upward linear trend.
This type of analysis shows up whenever researchers want to know whether one measurable quantity predicts another: fertiliser amount and crop yield, study hours and exam scores, advertising spend and revenue. The residual tells you how far any individual case departs from the overall trend, which is how analysts flag outliers or unusual observations in practice.
Students often describe the association without naming both variables. "There is a positive linear association" earns no context credit on the AP exam. You must say what the variables are.
Students sometimes compute residuals as ŷ – y instead of y – ŷ. The convention is always observed minus predicted; reversing it flips the sign and leads to a wrong interpretation.
Students frequently interpret slope without saying "predicted" or "on average." The regression line gives predicted values, not guarantees for every individual observation.
Students confuse "same sample size" with "same variability." Adding different flower types adds a new source of scatter regardless of how many flowers are in the study.
⚠️ Describing an association is one of the most common free-response tasks. The rubric typically requires all four elements: direction, form, strength, context. Leaving one out drops you a scoring level.
⚠️ Slope interpretation must include "predicted" (or equivalent language), the numerical change with units, and both variable names in context. Generic phrasing like "y goes up by 15.8 for each unit of x" is not sufficient.
⚠️ Residual = observed – predicted. The sign matters. Getting the sign wrong means your interpretation says the opposite of what happened.
⚠️ r² questions that introduce a new source of variability almost always have the answer "r² decreases." The reasoning must explain where the new variability comes from and why the linear model cannot account for it.
True or false: A scatterplot description that says "positive, linear, strong" but does not name the variables would earn full credit on an AP free-response question.
Fill in the blank: The slope of the LSRL tells you the change in the __________ value of y for each one-unit increase in x.
True or false: A residual of –5 means the observed value was 5 units higher than the model predicted.
Fill in the blank: r² measures the proportion of __________ in the response variable that is explained by the linear relationship with the explanatory variable.
True or false: If you add a new source of variability to an experiment (e.g. mixing flower types), r² will likely increase.
(Answers: 1. False, context is required. 2. predicted. 3. False, it was 5 units lower. 4. variability (or variation). 5. False, r² will likely decrease.)
Q: Describe the association shown in a scatterplot where points rise from left to right in a roughly straight line, tightly clustered, for the relationship between study hours and exam scores.
A: There is a strong, positive, linear association between the number of hours studied and exam score. As study hours increase, exam scores tend to increase as well.
Q: The LSRL for a dataset is ŷ = 180.8 + 15.8x, where x is tablespoons of sugar and ŷ is predicted hours of freshness. Interpret the slope.
A: For each additional tablespoon of sugar added to the vase, the predicted number of hours the carnation stays fresh increases by 15.8 hours.
Q: A carnation in a vase with 2 tablespoons of sugar actually stayed fresh for 204 hours. The LSRL predicts 212.4 hours. Calculate and interpret the residual.
A: Residual = 204 – 212.4 = –8.4 hours. The carnation stayed fresh for 8.4 fewer hours than the regression model predicted.
Q: A second experiment uses 12 flowers of mixed varieties instead of only carnations, with the same sugar levels. Would you expect r² to be higher, lower, or about the same compared to the carnation-only experiment? Explain.
A: Lower. Different flower varieties introduce additional variability in freshness that is unrelated to sugar amount. Because the linear model can only account for variation explained by sugar, the extra scatter from mixing varieties means a smaller proportion of total variation is explained, so r² decreases.
This material connects directly to residual plots (Ch. 3), which you use to check whether a linear model is appropriate. If the residuals show a pattern, the relationship may be non-linear, and you would consider a transformation.
It also sets the stage for inference for regression (later in the course), where you test whether the true population slope is significantly different from zero. Understanding what slope and residuals mean now makes the hypothesis-testing framework much easier to follow later.
scatterplot, scatter plot, bivariate data, explanatory variable, response variable, least-squares regression, LSRL, line of best fit, slope interpretation, y-intercept, residual, prediction error, observed minus predicted, r-squared, coefficient of determination, proportion of variation explained, AP Statistics Chapter 3, FRAPPY, carnation freshness experiment, sugar and freshness, describing associations, direction form strength context