Exploring One-Variable Data and Collecting Data
Unit 1 of AP Statistics, worth 20–30% of the exam. 14 questions below, each with the working. Every answer was checked by a second pass before it was published.
Categorical and quantitative variables, graphical and numerical summaries, normal distributions, sampling methods, bias, experimental design.
How this unit is tested
Start every problem by identifying the type of variable involved: categorical variables name groups (even if labeled with numbers, like zip codes), while quantitative variables are counts or measurements that can be meaningfully averaged (further split into discrete and continuous). That decision determines which graph applies — bar chart or pie chart for categorical data; dotplot, stemplot, or histogram for quantitative data — and which numerical summary is appropriate: mean and standard deviation for roughly symmetric data, median and IQR when the distribution is skewed or has outliers.
When a problem says a distribution is approximately normal, first try the 68-95-99.7 Empirical Rule for quick estimates at whole-number standard deviations from the mean. For any other value, convert to a z-score using $z=(x-\mu)/\sigma$ and use a standard normal table or calculator (normalcdf, invNorm) to find exact proportions or percentiles. Always sketch the curve and shade the region asked for before calculating — this catches sign and direction errors.
For data-collection questions, first decide whether the study is observational or an experiment. Observational studies only describe association and are vulnerable to confounding variables; only a well-designed experiment with random assignment can support a cause-and-effect conclusion. In experiment questions, identify the treatments, experimental units, response variable, and any control group, placebo, or blinding, and check whether blocking was used to control a known source of variability.
For sampling questions, identify the population, the sampling frame, and the specific method used — simple random sample, stratified, cluster, systematic, convenience, or voluntary response — then evaluate whether the method could produce bias (undercoverage, nonresponse, or response bias) and in which direction that bias would push the results.
What you have to know
Empirical Rule (68-95-99.7 Rule)
For a distribution that is approximately normal, about 68% of values lie within 1 standard deviation of the mean, about 95% lie within 2 standard deviations, and about 99.7% lie within 3 standard deviations.
Z-score formula
$z=\dfrac{x-\mu}{\sigma}$ converts a value $x$ into the number of standard deviations it lies above (positive) or below (negative) the mean, allowing comparison across different normal distributions.
1.5×IQR Outlier Rule
A value is flagged as an outlier if it is below $Q_1-1.5\times IQR$ or above $Q_3+1.5\times IQR$, where $IQR=Q_3-Q_1$.
Sample Standard Deviation
$s=\sqrt{\dfrac{\sum (x_i-\bar{x})^2}{n-1}}$; dividing by $n-1$ rather than $n$ corrects for the tendency of sample data to underestimate population variability.
Causal Conclusions Require Random Assignment
Observational studies, no matter how large, can only establish association because of possible confounding variables. Only a randomized experiment, in which subjects are randomly assigned to treatments, can support a cause-and-effect conclusion.
14 practice questions
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A city collects data on residents' zip codes to determine service districts. Even though it is recorded as a number, zip code is best classified as what type of variable?
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Answer. Categorical variable
Zip codes are labels that identify a location or group; averaging them would be meaningless, which is the defining test for a categorical variable regardless of its numeric appearance. -
The number of pets owned by each student in a class is recorded. What type of variable is this, and why?
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Answer. Quantitative discrete, because it is a countable numeric value (0, 1, 2, ...) with meaningful arithmetic, and it can only take whole-number values.
Quantitative variables measure or count something; since the number of pets is a whole-number count rather than a measurement on a continuous scale, it is discrete rather than continuous. -
Exam scores are approximately normal with mean 75 and standard deviation 5. According to the Empirical Rule, what percent of scores fall between 65 and 85?
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Answer. 95%
65 and 85 are each exactly 2 standard deviations from the mean (75 ± 2×5), and the Empirical Rule states about 95% of values in a normal distribution fall within 2 standard deviations of the mean. -
A data set of package weights is approximately normal with mean 50 g and standard deviation 4 g. What proportion of packages weigh above 58 g?
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Answer. About 2.5%
58 g is exactly 2 standard deviations above the mean (50 + 2×4 = 58). By the Empirical Rule, about 95% of values lie within 2 SD, leaving 5% in the two tails combined, or about 2.5% in each tail. -
Package weights are approximately normal with mean 12 oz and standard deviation 0.5 oz. Find the z-score for a package weighing 12.8 oz and interpret it in context.
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Answer. z = 1.6; the package weighs 1.6 standard deviations above the mean weight of 12 oz.
Using $z=(x-\mu)/\sigma$: $z=(12.8-12)/0.5=1.6$. A positive z-score of 1.6 means the value is well above average but within a plausible range for a normal distribution. -
A data set has Q1 = 20 and Q3 = 32. Using the 1.5×IQR rule, is a value of 51 an outlier?
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Answer. Yes, 51 is an outlier.
IQR = 32 − 20 = 12, so the upper fence is $Q3+1.5\times IQR = 32+18=50$. Since 51 exceeds 50, it is flagged as a high outlier. -
In a distribution, the mean is noticeably greater than the median. What does this suggest about the shape, and which measure of center should be reported?
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Answer. The distribution is likely right-skewed (or affected by high outliers); the median should be reported since it is resistant to skew and extreme values, unlike the mean.
A long right tail pulls the mean upward more than the median, producing mean > median. Because the mean is sensitive to skew and outliers, the median gives a more representative 'typical value' in this case. -
A school administrator randomly selects 5 entire classrooms out of 40 and surveys every student in those classrooms about cafeteria food. What sampling method is this?
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Answer. Cluster sample
In cluster sampling, naturally occurring groups (here, classrooms) are randomly selected as whole units, and every individual within each chosen cluster is surveyed — unlike stratified sampling, which samples from every group. -
A pollster divides voters into Democrat, Republican, and Independent groups, then takes a separate simple random sample from each group proportional to its size. What sampling method is this, and why might it be preferred over a plain SRS?
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Answer. Stratified random sampling; it guarantees representation from each political group and can reduce variability in the estimate because each stratum is more homogeneous with respect to the variable of interest.
Stratified sampling divides the population into meaningful subgroups first, then samples within each, which controls for a known source of variation that an SRS might handle unevenly by chance. -
A survey asks: 'Don't you agree that the wasteful spending in the new city budget should be reduced?' What type of bias does this introduce?
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Answer. Response bias, specifically wording bias.
The loaded, leading phrasing ('wasteful,' 'don't you agree') pushes respondents toward a particular answer regardless of their true opinion, systematically skewing the results. -
A radio show asks listeners to call in and vote on a proposed tax; only 200 of thousands of listeners call in. What kind of sample is this, and what direction of bias is likely?
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Answer. A voluntary response sample; it is likely biased toward people with strong opinions (often those opposed to the status quo), since participation is self-selected rather than random.
Voluntary response samples systematically overrepresent people motivated enough to respond on their own, which usually skews results away from the true population opinion. -
A study finds that people who take vitamin supplements have lower rates of heart disease than people who don't, and researchers conclude vitamins prevent heart disease. What is the flaw, and what term describes the underlying issue?
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Answer. This is an observational study, not a randomized experiment, so it cannot establish causation; a confounding variable such as overall health consciousness could explain both higher supplement use and lower heart disease rates.
Without random assignment, groups may differ systematically in ways related to the outcome, so any observed association could be due to a lurking or confounding variable rather than the supplement itself. -
In a randomized controlled drug trial, why do researchers randomly assign subjects to treatment and control groups rather than letting subjects choose?
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Answer. Random assignment balances known and unknown confounding variables between the groups on average, so any difference in outcomes can be attributed to the treatment, supporting a causal conclusion.
If subjects self-selected their group, factors like age, health, or motivation could differ systematically between groups and confound the results; randomization spreads these factors evenly by chance. -
A clinical trial gives one group a real medication and another an identical-looking sugar pill, and neither the subjects nor the evaluating researchers know who received which. What is this design called, and what does it control for?
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Answer. A double-blind, placebo-controlled experiment; it controls for the placebo effect (subject expectations) and for evaluator bias in measuring or interpreting outcomes.
Blinding subjects prevents their expectations from influencing the response, and blinding the evaluators prevents unconscious bias in recording or interpreting results, isolating the true effect of the treatment.
What people get wrong
- Dividing by n instead of n-1 when computing a sample standard deviation by hand. Instead, use $n-1$ (or trust the calculator's Sx, not σx, for sample data) since dividing by n underestimates spread.
- Concluding causation from an observational study (e.g., 'vitamin use causes lower heart disease rates'). Instead, state only that the variables are associated and note a plausible confounding variable, since causal claims require random assignment.
- Confusing stratified and cluster sampling. Instead, remember that stratified sampling takes a random sample from every group (each stratum is represented), while cluster sampling randomly selects a few whole groups and surveys everyone within them.
- Reporting the mean and standard deviation for a skewed distribution or one with outliers. Instead, check the shape first (from a graph or the mean-vs-median comparison) and report the median and IQR when the data are not roughly symmetric.
- Applying z-scores or the Empirical Rule without confirming the distribution is approximately normal. Instead, verify the problem states normality (or check a graph for a roughly symmetric, bell-shaped, single-peaked pattern) before using these tools.
- Treating random sampling and random assignment as the same thing. Instead, remember random sampling lets you generalize results to a population, while random assignment lets you make cause-and-effect claims about treatments; a study needs the specific type of randomness that matches the claim being made.
Drill this unit until it sticks
These questions come back on a schedule built from what you get wrong, alongside the rest of AP Statistics. Free, and no account needed to start.