1.2 The Language of Variation: Variables

Look at what the values mean, not just their form. - Categorical (qualitative): values are labels or group names—they sort individuals into categories. Examples: dominant hand, highest degree, age group (young/old). They can be nominal (no order) or ordinal (ordered). You count frequencies and use bar/two-way tables or chi-square tests on the AP exam (CED VAR-1.C.1). - Quantitative: values are numbers that measure or count something you can do arithmetic with (mean, SD). Examples: age in years, height, concentration. Quantitative variables are discrete (countable, e.g., number of pets) or continuous (measured, e.g., weight). You describe distributions with histograms, dotplots, mean/median, and use t/z procedures when appropriate (CED VAR-1.C.2). Quick test: can you sensibly compute an average that answers a real question? If yes → quantitative. If the “numbers” are really codes (like 1 = male, 2 = female) or labels → categorical. For more examples and practice, see the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt), the Unit 1 overview (https://library.fiveable.me/ap-statistics/unit-1), and practice problems (https://library.fiveable.me/practice/ap-statistics).

Categorical vs quantitative: simple rule—categorical (qualitative) variables label groups or categories; quantitative variables give numeric measurements you can do math with. CED terms: categorical values are category names or group labels (nominal like “dominant hand” or ordinal like “age group: young/old”); quantitative values are numbers measuring/counting something (discrete like “number of siblings” or continuous like “height” or “age of a structure”). Why it matters for AP Stats: the type of variable decides how you display and analyze data (counts/proportions, bar/pareto charts for categorical; histograms, stem-and-leaf, mean/SD for quantitative). On the exam you’ll need to identify variables and pick appropriate summaries/tests (Skill 2.A; Topic 1.2 in the CED). Want quick practice? Review the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and try problems at (https://library.fiveable.me/practice/ap-statistics).

A variable is any characteristic that can change from one individual (observational unit) to another—it’s what you measure or record. Values are the specific outcomes for each individual. The AP CED splits variables into two types: categorical (labels/groups like dominant hand, age group, highest degree) and quantitative (numeric measures like height, age, concentration). Categorical can be nominal or ordinal; some are binary (two categories). Quantitative can be discrete (countable) or continuous (measured). In studies you’ll also hear explanatory vs. response variables: explanatory predicts or explains, response is the outcome. Remember measurement error and natural variability make conclusions uncertain (VAR-1). You’ll need to identify and classify variables on the exam (Topic 1.2) and use that to choose graphs/tests. For a quick review, see the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Classify a variable by what its values represent. If values are names or group labels (colors, majors, yes/no, age group, highest degree) it’s categorical—think nominal or ordinal (order matters only for ordinal). If values are numbers that measure or count something (age in years, height, concentration, # of pets) it’s quantitative—either discrete (counts) or continuous (measurements). Quick test: can you reasonably average or take a difference of the values? If yes, it’s quantitative. Also note the observational unit is the individual you measure (people, plots, samples). This distinction is in the CED (VAR-1.C.1 and VAR-1.C.2) and is tested in Unit 1 on the AP exam. For a focused review, check the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and practice problems (https://library.fiveable.me/practice/ap-statistics).

Short answer: age can be either. If you record exact years (or days, months)—e.g., 17, 17.5, 65—age is quantitative (measured/counted). It can be treated as discrete (whole years) or continuous (exact age in years). That matches the CED definition: quantitative = numerical values for a measured or counted quantity (VAR-1.C.2). But if you put people into groups like “child/teen/adult,” “young/old,” or bins (0–17, 18–64, 65+), age is categorical (often ordinal) because values are labels, not raw numbers (VAR-1.C.1). On the AP exam you’ll be asked to identify which type a variable is (Skill 2.A), so pay attention to how the data are recorded—grouped vs. measured. Want more examples and quick practice on variable types? Check the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and more practice problems (https://library.fiveable.me/practice/ap-statistics).

Start by finding the individuals (observational units)—the “who” or “what” each row describes. A variable is any characteristic that varies across those individuals (CED: VAR-1.B.1). Then classify each variable: - Categorical: labels or groups (nominal = names like dominant hand; ordinal = ordered groups like age group; binary/dichotomous = two categories). - Quantitative: numeric measurements or counts (discrete = countable, like # of pets; continuous = measurements, like height or concentration). Quick checklist for the AP exam: write the observational unit, name the variable, say whether it’s categorical or quantitative, and give type (nominal/ordinal or discrete/continuous). On FRQs you’ll often need that plus context and units. Unit 1 is tested heavily (15–23% of the MC section), so practice identifying variables until it’s automatic. For targeted review see the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and try lots of problems at (https://library.fiveable.me/practice/ap-statistics).

Short rule: ask three quick questions. 1) Are the values labels or group names (words, colors, yes/no)? If yes → categorical (nominal if no order, ordinal if there’s a meaningful order). 2) Are the values numbers that you can add, average, or take differences of? If yes → quantitative. 3) If quantitative, is it counting whole items (0,1,2,...) → discrete, or measuring on a continuum (height, time, concentration) → continuous. Examples: “Highest degree earned” = categorical (nominal/ordinal), “age group (young/old)” = categorical (ordinal), “height of a child” = quantitative continuous, “number of pets” = quantitative discrete, “dominant hand” = categorical binary/dichotomous. For AP Stats (Topic 1.2 VAR-1.C), you should be able to identify variables and label them using these terms. Review the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and drill practice (https://library.fiveable.me/practice/ap-statistics).

Step-by-step to classify a variable (quick guide you can use on the exam): 1. Identify the observational unit and variable value (who/what and what’s measured). 2. Ask: is the value a number that measures/counts something? If yes → quantitative. If it’s a label/name/group → categorical. (CED: VAR-1.C.1 & VAR-1.C.2) 3. For categorical: decide nominal (no order, e.g., dominant hand) or ordinal (has order, e.g., age group). Note binary/dichotomous if only two categories. 4. For quantitative: decide discrete (countable, e.g., # of pets) or continuous (measured, can take many values, e.g., height). 5. Put variable in context: is it an explanatory or response variable for the study? (helps with later analysis) 6. Check for measurement error or weird values that might affect classification. 7. Practice: classify variables from past AP prompts—this shows up early in Unit 1 on the exam. For a topic review, see the Fiveable study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and try practice problems (https://library.fiveable.me/practice/ap-statistics).

They're not the same in how you record and analyze them. "Age" is quantitative because you measure it with numbers (years, months) and can do arithmetic—means, standard deviations, histograms, etc. The CED defines a quantitative variable as numeric for a measured or counted quantity (VAR-1.C.2). "Age group" is categorical: you place each person into a label (young, middle, old or 0–9, 10–19, …). Those are category names or group labels (VAR-1.C.1). Often age groups are ordinal (they have a natural order) but they’re still categorical because you’ve turned numbers into labels and lost the exact numeric values—so you would use bar charts or two-way tables, not a histogram or mean of the labels. Which form you use matters on the AP exam: pick methods appropriate to the variable type (graphs and summaries for quantitative vs. categorical). For a quick review, see the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt). For more practice, check Unit 1 (https://library.fiveable.me/ap-statistics/unit-1) and the practice question bank (https://library.fiveable.me/practice/ap-statistics).

Start by asking: who/what are the individuals (observational units)? A variable is any characteristic that can vary across those individuals (VAR-1.B.1). To identify all variables in a problem, list every measured or recorded characteristic (e.g., age, gender, test score, treatment group, time). Then classify each variable: - Categorical (names/labels): nominal (e.g., dominant hand), ordinal (e.g., small/medium/large), binary/dichotomous (yes/no). (VAR-1.C.1) - Quantitative (numbers you measure or count): discrete (counts) or continuous (measurements like height, concentration). (VAR-1.C.2) Also note roles when relevant: is a variable an explanatory or a response variable? And check units and possible measurement error. On the AP exam you’ll be asked to identify variables and their type (Skill 2.A, Topic 1.2), so practice by reading prompts, listing individuals, then writing each variable name + type. For a quick refresher use the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and drill practice problems at (https://library.fiveable.me/practice/ap-statistics).

Memorize a handful of clear categorical examples and their types so you can spot them fast on the exam. Categorical variables (VAR-1.C.1) take names or labels—think nominal, ordinal, or binary/dichotomous. Useful examples to remember: - Nominal: eye color, blood type, dominant hand, favorite music genre - Ordinal: education level (high school / bachelor’s / grad), pain scale (none / mild / severe), age group (young / middle / old) - Binary/dichotomous: sex (male/female), passed exam (yes/no), owns a car (yes/no) Quick tips: if values are labels or ranks, it’s categorical; if they’re measured numbers (height, age in years, concentration) it’s quantitative (VAR-1.C.2). The AP exam often asks you to identify variables or pick appropriate methods—Unit 1 questions make up ~15–23% of the multiple-choice (see CED). Review this Topic 1.2 study guide for more examples (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and practice lots of classification problems (https://library.fiveable.me/practice/ap-statistics).

Income is usually a quantitative variable because it takes numerical values that measure an amount (dollars earned)—that fits the CED definition: a quantitative variable “takes on numerical values for a measured or counted quantity” (VAR-1.C.2). Income can be treated as continuous (any dollar/cent value) or as discrete if you record whole dollars or counts, but either way it’s numeric and you can compute means, SDs, etc. Important caveat: if you record income as brackets (e.g., $0–$25k, $25–$50k), you’ve turned it into a categorical (ordinal) variable because you’re recording group labels instead of exact numbers. On the AP exam you should identify income as quantitative unless the problem explicitly says it’s grouped into categories (see Topic 1.2 study guide for examples: https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt). For extra practice, check more problems at (https://library.fiveable.me/practice/ap-statistics).

Scan the problem for the things that change from one person/thing to another—those are your variables. Ask: who/what are the individuals (observational units)? What characteristics were measured or recorded for each individual? Those characteristics = variables. Then classify each variable: is it a category/label (categorical: nominal, ordinal, binary) or a number you can measure/count (quantitative: discrete or continuous)? Quick checklist you can use on every word problem: - Identify the individuals (people, cities, trials). - For each sentence that describes “what we record,” ask “does this vary?” If yes, it’s a variable. - Look at values: words/labels → categorical; numbers with units → quantitative. - Note any explanatory vs response role only if the question asks about relationships. Practice spotting variables with problems in the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and try many practice questions (https://library.fiveable.me/practice/ap-statistics)—that builds pattern recognition for the AP exam (Skill 2.A).

It matters because the type of variable tells you what summaries, graphs, and inference methods are appropriate—and the AP exam checks that. Categorical variables (labels, groups) are summarized with counts/proportions and shown with bar/pareto charts or contingency tables; quantitative variables (numbers) use center/shape/spread (mean & SD or median & IQR) and histograms/boxplots. For inference, categorical outcomes lead to tests about proportions or chi-square; quantitative outcomes lead to means, t-tests, or regression. Using the wrong type (e.g., taking a mean of nominal labels) produces meaningless results and violates conditions for inference (randomness, independence, sample size). Topic 1.2 in the CED emphasizes identifying and classifying variables (VAR-1.C)—review it in the Fiveable study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt). For practice turning these ideas into exam-ready skills, try problems at Fiveable’s practice page (https://library.fiveable.me/practice/ap-statistics).

Short tricks that actually work: categorical = names/labels, quantitative = numbers you can measure or average. Ask yourself two quick questions for any variable: 1) Is it a label or a number? If label → categorical (nominal if no order, ordinal if order matters like “small/medium/large”, binary = 2 categories). 2) If it’s numeric, can you do arithmetic that makes sense (mean, SD)? If yes → quantitative. Then ask: is it a count (discrete: 0,1,2,...) or a measurement on a scale (continuous: can take any value between)? Extras for AP: identify the observational unit (individual) and whether a variable is explanatory (predictor) or response (outcome)—that matters on questions about study design. AP expects you to classify variables (VAR-1.B, VAR-1.C) and use correct terms like nominal/ordinal, discrete/continuous. Practice this on 10 examples (names vs numbers, then count vs measure). For a quick review, see the Topic 1.2 study guide (https://library.fiveable.me/ap-statistics/unit-1/language-variation-variables/study-guide/nKpeaxi1H3Ht9aFhTHKt) and try practice problems at (https://library.fiveable.me/practice/ap-statistics).