Types of Data and Sampling

Types of Data

Categorical (Qualitative) Data

• Data that describes qualities or characteristics
• Cannot be ordered meaningfully (not about size/amount)
• Examples: color, political party, type of fruit
• Divided into:
  • Nominal: no natural ordering (red/blue/green)
  • Ordinal: natural ordering (small/medium/large, freshman/sophomore/junior/senior)

Quantitative (Numerical) Data

• Data that represents measurements or amounts
• Can be ordered and arithmetic makes sense
• Examples: height, weight, age, test score
• Divided into:
  • Discrete: counts, whole numbers (number of siblings, cars sold)
  • Continuous: measured, any value in a range (height, time, weight)

Key Definitions

Population: entire group of individuals we want information about

• Example: all AP Statistics students in the United States

Sample: subset of the population we actually collect data from

• Example: 500 randomly selected AP Statistics students

Parameter: numerical summary of a population (unknown, fixed)

• Notation: usually Greek letters (μ, σ, p)
• Example: true mean score of all AP Stat students

Statistic: numerical summary of a sample (known, varies sample to sample)

• Notation: usually letters (\(ar{x}\), s, \(hat{p}\))
• Example: mean score of 500 sampled students

Sampling Methods

Simple Random Sample (SRS)

• Every individual has equal chance of selection
• Best method when population is accessible
• Use random number generator or table
• Eliminates selection bias

Stratified Random Sample

• Divide population into homogeneous groups (strata)
• Randomly sample from each stratum proportionally
• Better representation of subgroups
• Example: stratify by grade level, then randomly select from each

Cluster Sample

• Divide population into clusters (typically geographic)
• Randomly select entire clusters
• Less expensive than SRS; useful when population spread out
• Risk: clusters not representative

Systematic Sample

• Select every \(k\)-th individual from ordered list
• Simple to implement
• Risk: hidden pattern in list

Convenience Sample (avoid for inference)

• Sample whoever is easiest to reach
• Biased; generally not representative

Common Sampling Biases

Sampling bias (selection bias): certain individuals more likely to be selected

• Convenience sample at mall (misses online shoppers)

Response bias: individuals respond untruthfully or refuse

• Loaded question: "Don't you agree this policy is wasteful?"
• Shy respondents not answering honestly

Non-response bias: some selected individuals don't respond

• Mail survey with 40% return rate

Undercoverage: some part of population not accessible

• Phone survey (misses homeless)

Worked Example

Scenario: A school wants to estimate mean SAT score for all 1,200 seniors.

Method 1 (SRS): Generate 100 random student IDs from 1–1200, compute mean score for those students.

Method 2 (Stratified): Divide into 3 strata by gender (400 male, 500 female, 300 nonbinary). From each stratum, randomly select 33–34 students. Compute mean.

Method 3 (Systematic): Generate random starting point (say, 5), then select students 5, 17, 29, 41, ... until 100 selected.

Decision Rule: When to Use Each

• SRS: population list available, want unbiased sample, resources sufficient
• Stratified: important subgroups exist, want equal representation of subgroups
• Cluster: population geographically spread, population list unavailable, budget limited
• Avoid convenience/systematic: if inference accuracy is important

AP Exam Tip

On FRQ prompt about study design, identify:

1. Population (who are we studying?)
2. Sample method (how were subjects selected?)
3. Bias present? (selection, response, non-response, undercoverage?)
4. Why this method? (explain trade-offs)

Common error: assuming \(ar{x}\) = μ just because you have a large sample. Sampling bias can produce bad estimates even with large n.