AP Statistics · Lesson 1

Variables, questions and distributions

Know Know it

A statistical question is one you answer with data that you expect to vary. How long do students at one school spend on homework each night is statistical, because the answer differs from student to student. How long one student spent last night is not.

A variable is a characteristic recorded for each individual. A categorical variable puts individuals into groups, such as blood type or favorite sport. A quantitative variable takes number values that count or measure, such as height or the number of siblings, so averaging them makes sense. A quantitative variable is discrete if it takes separate countable values and continuous if it can take any value in an interval.

Show a categorical variable with a bar chart or pie chart, using counts or relative frequencies (count divided by total). Show a quantitative variable with a dotplot, stemplot or histogram.

Describe a quantitative distribution with four features, always in context: shape (roughly symmetric, skewed right with a long tail toward high values, skewed left, unimodal or bimodal), center (a typical value), variability (how spread out the values are) and unusual features (outliers, gaps or clusters).

One fictional class of 30 students: how each student gets to school
Travel modeCountRelative frequency
Bus120.40
Car90.30
Walk60.20
Bike30.10

Trap A number is not always quantitative

Zip codes, jersey numbers and phone area codes are written as digits, but they only label groups. Averaging them means nothing, so they are categorical. Ask whether arithmetic on the values makes sense.

Worked example Easy

A fictional blood drive records four facts about each donor. Which one is a categorical variable?

  1. The donor's blood typeAnswer
  2. The donor's height in centimeters
  3. The donor's number of past donations
  4. The donor's resting pulse in beats per minute

How to solve it

  1. Ask of each variable: does it put the donor in a group, or is it a number you could average?
  2. Blood type names a group; the other three are numbers that count or measure.

Why each choice is right or wrong

  • A. Correct. Blood type sorts each donor into a group such as A, B, AB or O, so it is categorical.
  • B. Incorrect. Height in centimeters is a measurement, so averaging heights makes sense and the variable is quantitative.
  • C. Incorrect. The number of past donations is a count, which makes it a quantitative (discrete) variable.
  • D. Incorrect. Resting pulse in beats per minute is a measured number, so it is quantitative, not categorical.

Explain Explain it

Task verbs: identify, describe

To describe a distribution, name all four features in context, and back up center, variability and any unusual values with numbers. Weak: the data are skewed with an outlier. Strong: commute times are skewed right, with a median of about 15 minutes and the middle half of students between 11 and 22 minutes, and one student's 58-minute commute stands far above the rest.

Worked example Medium

Fictional commute times of 52 workers, in minutes

Commute timeNumber of workers
0 to under 1014
10 to under 2022
20 to under 309
30 to under 404
40 to under 502
50 to under 601

What is the shape of this distribution of commute times?

  1. Skewed left, with a tail toward short commutes
  2. Roughly symmetric around 30 minutes
  3. Skewed right, with a tail toward long commutesAnswer
  4. Bimodal, with peaks at 10 and at 40 minutes

How to solve it

  1. Find the peak: the 10-to-20 interval holds 22 of the 52 commutes.
  2. Look at the tails: the counts drop slowly toward 60 minutes but stop sharply at 0.
  3. A long tail toward high values means skewed right.

Why each choice is right or wrong

  • A. Incorrect. Skewed left would put a long tail toward short commutes, but the counts thin out toward the long ones instead.
  • B. Incorrect. Roughly symmetric around 30 minutes does not fit, because only 4 of 52 commutes are between 30 and 40 minutes.
  • C. Correct. Most commutes fall below 20 minutes, and the counts shrink steadily toward 50 to 60 minutes, so the long tail is on the right.
  • D. Incorrect. Bimodal with peaks at 10 and 40 minutes does not fit, since the only peak is the 10-to-20-minute interval.

Identify Your turn

A fictional apartment complex records the number of pets living in each of its 40 apartments. Say whether this variable is categorical or quantitative and why, then say whether it is discrete or continuous and why.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • Says the variable is quantitative and gives a reason
  • Says the variable is discrete and gives a reason

Model answer

The number of pets is quantitative, because each value is a count and an average such as 1.2 pets per apartment makes sense. It is discrete, because an apartment can hold 0, 1, 2 or 3 pets but never 1.5 pets.

Apply Apply it

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