AP Statistics — exploring data

Describing one variable and then the relationship between two are the parts of AP Statistics that come before any probability: one-variable data sits in Unit 1 (Exploring One-Variable Data and Collecting Data, 20 to 30 percent of the multiple-choice section) and regression is Unit 5 (Regression Analysis, 10 to 20 percent). This set takes ten of their ideas, one question each. Every distribution and data set is described in words with the numbers written into the sentence; a histogram, a residual plot and a scatter plot with its fitted line are drawn as well.

Several questions are built so the answer can be computed rather than recalled. Nine ordered values have a tenth of 200 added to them: the median moves from 9 to 9.5 because it depends only on position, while the mean goes from 12 to 30.8 and the standard deviation rises sharply, because both depend on magnitude. That split is the whole of what resistance means, and seeing it in two arithmetic steps is more useful than the word. The fence question gives quartiles of 20 and 50, so the interquartile range is 30, one and a half of those is 45, and the fences sit at −25 and 95 — each measured outward from its own quartile; adding 45 to the lower quartile instead is the error that would wrongly flag a value of 60.

Three questions are about what a statistic does not say. The correlation coefficient measures linear association and nothing else, so points lying exactly on a parabola can give r near zero while the relationship is perfect — which is why r = 0 rules out a straight-line trend and rules out nothing else. The coefficient of determination reports a proportion of variation, not a probability that the model is right, and a model can post a high r² while being the wrong shape entirely. And a U-shaped residual plot means the line has missed something systematic; the tempting wrong answer, that the residuals balance out, is true of every least-squares fit by construction and therefore carries no information at all.

The last question is about extrapolation: a line fitted to saplings aged one to eight years says nothing reliable about a thirty-year-old tree, however well it fits inside the observed range.

Every explanation names which misunderstanding produces which wrong answer. Every data set, context and number here is invented and is attributed to no real study, institution or place, and nothing is drawn from any College Board publication, equations sheet or reference table.

Choose between the mean and the median for a skewed distribution, and state resistance as the reason
Predict how a single extreme observation moves the mean, median, standard deviation and interquartile range differently, and compute the change
Interpret a standard deviation in the units of the variable measured, against the mean
Compare values from two different distributions by standardizing, and explain why equal gaps above two means are not equal standing
Apply the $1.5 \times \text{IQR}$ rule outward from the quartiles to identify a flagged value
State what the correlation coefficient measures and the three things it does not — curvature, causation, and any dependence on units
Interpret a least-squares slope as a rate, and judge whether the intercept means anything in context
Read a residual plot, and explain why balanced residuals are guaranteed rather than informative
Interpret $r^2$ as a proportion of variation explained, distinguishing it from $r$ and from a probability
Recognize extrapolation and explain why a prediction outside the observed range of $x$ is unreliable

Built against the published structure of AP Statistics (Course and Exam Description effective fall 2026): Unit 1, Exploring One-Variable Data and Collecting Data, and Unit 5, Regression Analysis. The exam runs 3 hours and is taken digitally: 42 multiple-choice questions in 1 hour 30 minutes for 50 percent of the score, then four free-response questions in 1 hour 30 minutes for the remaining 50 percent, each worth ten points. A graphing calculator with statistical capabilities is expected, and an equations sheet and reference tables are supplied during the exam. Every data set, context and number in this material is invented, and none is attributed to a real study, institution or place. Nothing is reproduced from any College Board publication, released exam, course and exam description, scoring guideline, equations sheet or reference table. Zestly is an independent study tool. It is not affiliated with the College Board, which owns the AP Statistics exam, and it is not a test center.

Sample question

Histogram of house prices in thousands of dollars, from 100 to 800: most houses fall between 100 and 300, and a long thin tail stretches to the right.

House prices in a small village are strongly skewed to the right: most houses cluster together and a handful of very large properties sit far out in the upper tail. Which measure of center better describes what a typical house costs, and why?

The mean, because it uses every observation and so wastes no information
The median, because it depends on the position of the middle value rather than on the size of the extreme ones, and so is not dragged into the tail
The mean, because its sensitivity to the long tail makes it reflect the shape of the distribution more faithfully
The median, because in a skewed distribution the median and the mean coincide anyway

See the answer

The median, because it depends on the position of the middle value rather than on the size of the extreme ones, and so is not dragged into the tail

In a right-skewed distribution the mean sits above the median, pulled up by the far values, so it can exceed what most houses actually cost — a figure that is arithmetically correct and misleading as a description of the typical case. The median splits the ordered values in half and moves only when the middle moves, so a mansion twice as expensive shifts it not at all. That property is called resistance, and it is the reason for the choice. Using every observation is a genuine merit of the mean and exactly why it fails here: the extreme values are included at full weight. Calling that sensitivity an advantage confuses describing the SHAPE with describing the CENTER — the gap between mean and median is itself useful evidence of skew, but neither number is thereby a better summary of the typical value. And the mean and median coincide only in a symmetric distribution, which is precisely what this one is not.

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