AP® Statistics review sheet from Aim for Five (aimforfive.com/stats/units/1/1-2)
Unit 1 · Topic 1.2
1.2 Variables
Before you can analyze data, you have to know what was measured and on whom. This topic covers observational units, the two kinds of variables (categorical and quantitative) and the difference between a parameter and a statistic. These labels decide which graphs, summaries and inference methods you'll use later.
Key terms
- observational unit
- categorical variable
- quantitative variable
- discrete vs. continuous
- parameter
- statistic
Observational units and variables
An observational unit is the person or thing you collect data from. In a survey of students, each student is an observational unit. In a study of cereal boxes, each box is.
A variable is a characteristic that can change from one unit to the next: a student's grade level, a box's weight. Data aren't only numbers and words. Photos, sound clips, videos and text can all be data once you record something about each unit.
Categorical vs. quantitative
A categorical variable (also called qualitative) puts each unit into a group or category: eye color, phone brand, yes/no answers. A quantitative variable (also called numerical) records a measured or counted amount, usually with units: height in centimeters, number of siblings, price in dollars.
The test is whether arithmetic makes sense. Averaging heights gives a meaningful number. Averaging ZIP codes or jersey numbers doesn't, so those are categorical even though they're written with digits.
| Variable | Type | Why |
|---|---|---|
| Favorite sport | Categorical | Values are names of groups |
| ZIP code | Categorical | Digits are labels; averaging is meaningless |
| Number of pets | Quantitative, discrete | Counted: 0, 1, 2, … |
| Time to run a mile | Quantitative, continuous | Measured; any value in an interval is possible |
Discrete vs. continuous
Quantitative variables come in two kinds. A discrete variable takes values you can count and list, like 0, 1, 2, 3, and so on. The list might be finite (the number of heads in 10 coin flips) or go on forever (the number of emails you'll get this year has no fixed upper limit).
A continuous variable can take any value in an interval. Between 5.1 seconds and 5.2 seconds there are infinitely many possible times. Measurements of time, length, weight and temperature are continuous even if you round them when you record them.
Parameters and statistics
A parameter is a number that describes a population, such as the true proportion of all U.S. teens who own a smartphone. A statistic is a number calculated from a sample, such as the proportion of 1,000 surveyed teens who own one.
Memory trick: Population goes with Parameter, Sample goes with Statistic.
You usually can't know a parameter, so you use a statistic to estimate it. The statistic will almost never equal the parameter exactly, and it changes from sample to sample. Much of this course is about how far off a statistic is likely to be.
Common symbols: μ (mean) and σ (standard deviation) and p (proportion) are parameters; x̄, s and p̂ are the matching statistics.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Classify the variables
A vet clinic records, for each dog it treats: breed, weight in kilograms, number of visits this year and whether the dog is microchipped. Identify the observational units and classify each variable.
Show the solutionHide the solution
- Step 1: The observational units are the dogs the clinic treats, because each record describes one dog.
- Step 2: Breed is categorical: values are group names.
- Step 3: Weight is quantitative and continuous: it's measured and can be any value in a range.
- Step 4: Number of visits is quantitative and discrete: it's a count (0, 1, 2, …).
- Step 5: Microchipped (yes/no) is categorical.
Answer: Units: dogs treated at the clinic. Breed and microchipped: categorical. Weight: quantitative, continuous. Visits: quantitative, discrete.
- Example 2Calculator allowed
Parameter or statistic?
A school has 900 students. A counselor asks 75 randomly selected students how many hours they work at a job each week and finds a mean of 6.2 hours. The school's records show that 38% of all 900 students have a job. Is 6.2 a parameter or a statistic? What about 38%?
Show the solutionHide the solution
- Step 1: 6.2 hours comes from the 75 students in the sample, so it's a statistic. Its symbol is x̄ = 6.2.
- Step 2: 38% describes all 900 students, the entire population, so it's a parameter. Its symbol is p = 0.38.
Answer: 6.2 hours is a statistic (x̄); 38% is a parameter (p).
Common mistakes
- Calling a variable quantitative just because it's written with numbers. ZIP codes, phone numbers and jersey numbers are categorical.
- Mixing up the variable with its values. "Blue" is a value; "eye color" is the variable.
- Calling a measured variable discrete because the data were rounded. Rounded times are still continuous.
- Using x̄ or p̂ for a population value. Greek letters (μ, σ) and p are for populations.
On the exam
- Multiple-choice questions often ask which number in a scenario is a parameter or which variable is categorical. Ask: does it describe everyone, or just the sample? Does averaging it make sense?
- On free response, define the parameter in words with context, for example "p = the true proportion of all students at the school who have a job." You'll need this skill in every inference question.
Connected topics
Videos
Check yourself
4 questions on 1.2 Variables. Pick an answer to see if you got it, and why.
A hospital records the following for each newborn: birth weight (grams), number of siblings, blood type, and the zip code of the parents' home. How many of these four variables are categorical?
Which of the following variables is quantitative and continuous?
A biologist measures the length and mass of 40 trout caught in a lake and records the water temperature where each was caught. What are the observational units?
A principal wants to know what percent of the 1,850 students at her high school visited the school library at least once last month. She randomly selects 120 students from the enrollment list and asks each one. Of these, 46% say they visited the library at least once last month.
Described scenario
Which of the following is a statistic?
0 of 4 answered