AP Statistics: how to answer free-response questions and interpret computer output for full credit

What AP Statistics free-response actually demands

The free-response section is half the AP Statistics exam, and it is where students with sound statistical knowledge most often leave marks on the table, not because the statistics are wrong but because the communication is. The College Board scores free-response questions on four skill categories, and across all of them the recurring requirement is the same: answer in context, show your reasoning, and respect the limits of the data. This guide ties together the Unit 1 and Unit 2 dot-point pages, each with its own practice: describing the distribution of a quantitative variable, comparing distributions of a quantitative variable, summary statistics for a quantitative variable, correlation, least squares regression, and residuals.

The golden rule: answer in context

The most reliable way to gain (or lose) free-response marks is context. A statement that is statistically correct but generic, "the median is higher," "the slope is $0.6$," "there is a strong positive correlation," will often score below a statement that names the variables, units, and group. The scoring guidelines are written around context: a description of a distribution must say what is being measured and in what units; a slope interpretation must name both variables; a conclusion about association must reference the actual study.

Describing a distribution for full credit

Describing a single quantitative distribution is one of the most common free-response tasks. Cover shape, center, spread, and unusual features (SOCS), each in context, and match your measures to the shape.

• Shape: symmetric or skewed (named by the tail, not the hump), and the number of peaks (unimodal, bimodal).
• Outliers and unusual features: gaps, clusters, and isolated points; use the $1.5 \times \text{IQR}$ rule when asked to justify an outlier formally.
• Center: a typical value, the median (resistant) for skewed data or the mean for symmetric data.
• Spread: the IQR (resistant) for skewed data or the standard deviation for symmetric data, with the range as a last resort.

Markers award the components separately, so the fastest way to lose a point is to omit one, most often spread or unusual features. Run the SOCS checklist deliberately every time.

Comparing distributions: use comparative language

When a question asks you to compare distributions, two correct one-group descriptions placed side by side do not earn the comparison marks. You must use explicitly comparative language.

A reliable structure is four sentences, one each for shape, center, spread, and unusual features, every sentence naming both groups, the relevant measure, a comparison word, and the units.

Reading regression computer output

AP Statistics free-response questions frequently give a regression table from software rather than the equation directly, and expect you to extract and interpret it. A typical layout reads:

Predictor     Coef     SE Coef      T        P
Constant      12.4     2.1          5.90     0.000
HoursStudied   8.0     0.9          8.89     0.000

S = 4.2     R-Sq = 81.0%     R-Sq(adj) = 80.2%

Here is how to read it for full credit:

• Intercept (Constant row, Coef): $12.4$. Build the equation as $\hat{y} = 12.4 + 8.0x$.
• Slope (predictor row, Coef): $8.0$, the coefficient on HoursStudied. Interpret: "for each additional hour studied, the predicted exam score increases by $8.0$ points, on average."
• $s$ (the S value): $4.2$, the standard deviation of the residuals. Interpret: "predicted exam scores are typically off by about $4.2$ points."
• $r^2$ (R-Sq): $81.0\%$. Interpret: "about $81\%$ of the variation in exam score is explained by the linear relationship with hours studied."
• $r$ (correlation): $\sqrt{0.81} = 0.9$, taking the sign of the slope (positive here), so $r = 0.9$.

The three cautions that protect your marks

Three limits of the data come up in almost every Unit 1 to Unit 2 free-response question, and respecting them is part of full-credit reasoning.

1. Correlation is not causation. An association in observational data may be driven by a lurking variable, by reverse causation, or by coincidence. Only a randomised experiment supports a causal claim. When you find an association, describe it and then decline to claim cause, naming a plausible lurking variable if asked.
2. The normal model needs approximately normal data. The empirical ($68$-$95$-$99.7$) rule and z-score proportions only hold for roughly symmetric, bell-shaped data. If a distribution is clearly skewed, do not reach for the normal model.
3. Extrapolation is unreliable. A regression line is trustworthy only within the range of the observed data. Predicting outside that range assumes a pattern you have not seen, and can produce nonsense (such as a negative price). Compute if asked, but flag it as an extrapolation.

Showing work and justifying conclusions

The calculator does the arithmetic, but the marks are in the reasoning. When you describe an outlier, show the $1.5 \times \text{IQR}$ fences. When you judge whether a linear model fits, cite the residual plot (random scatter means it fits; a curve means it does not), not just a high $r^2$. When you conclude an association from a two-way table, compare the conditional distributions, not raw counts. A bare correct answer with no supporting reasoning frequently scores below a well-justified answer, because the skill categories explicitly reward justification. Treat every "explain" or "justify" as a request to make your statistical reasoning visible, in context.

Check your knowledge

A mix of description, comparison, regression-output, and caution questions. Write full-credit, in-context answers, then check against the quiz solutions.

1. Rewrite "the mean is $50$ and the standard deviation is $10$" as a contextual description for a distribution of resting heart rates (bpm).
2. A distribution of incomes is strongly right-skewed. Which center and spread should you report, and why?
3. Turn "Group A median $30$, Group B median $24$" into a proper comparison.
4. From output with Constant Coef $= 5$ and slope Coef $= 2$ (predicting $y$ from $x$), write the equation and interpret the slope.
5. R-Sq $= 49\%$ in a regression of $y$ on $x$. Interpret $r^2$ and find $r$ if the slope is negative.
6. A study finds an association between coffee drinking and heart disease in observational data. What can and cannot be concluded?
7. A regression valid for $x$ from $5$ to $20$ is used to predict at $x = 40$. What is the problem?
8. A residual plot of a linear fit shows a clear U-shape. Is the linear model appropriate? Justify.

For the official guidance

The College Board publishes released free-response questions and scoring guidelines at apcentral.collegeboard.org. Studying the official scoring guidelines shows exactly how context, comparative language, and justification are rewarded, so always practice against the board's own released exams.