📖 抽样方法：简单随机、分层、系统

Quantitative Methods — Sampling and Estimation · Lesson 1

1. Why Sampling?

1.1 Population vs Sample

💡 Core Logic: Because surveying an entire population is too costly or impossible (e.g., destructive testing), we use sample statistics to estimate population parameters. This forms the foundation of Statistical Inference.

1.2 Common CFA Scenarios

• A fund manager wants to assess the overall profitability of a sector → cannot analyze every company → takes a sample
• Quality control tests product lifespan → cannot destroy every unit → samples for testing
• An analyst surveys consumer confidence → cannot interview the entire nation → uses sampling

2. Classification of Sampling Methods

Sampling Methods
│
├── Probability Sampling
│   ├── Simple Random Sampling
│   ├── Stratified Random Sampling
│   └── Systematic Sampling
│
└── Non-Probability Sampling
    ├── Convenience Sampling
    ├── Judgmental Sampling
    └── Quota Sampling

📌 CFA Level 1 focuses heavily on the three probability sampling methods and their pros/cons. Non-probability sampling only requires knowing the definitions and limitations.

3. Simple Random Sampling

3.1 Definition

Drawing a sample from a population such that every individual has an equal probability of being selected, and each draw is independent of others.

3.2 Procedure

• Assign a unique number to each individual in the population (1 to N)
• Use a random number generator (Excel RAND(), Python random.sample()) to select n numbers
• The individuals corresponding to those numbers form the sample

3.3 Practical Example

Case: An analyst wants to randomly select 50 stocks from the S&P 500 for financial ratio analysis.

• Number all 500 stocks from 1 to 500
• Use =RANDBETWEEN(1,500) to generate 50 unique random numbers
• The selected 50 stocks form the sample

Probability of any individual stock being selected = 50/500 = 10%

3.4 Advantages

3.5 Disadvantages

4. Stratified Random Sampling

4.1 Definition

First divide the population into mutually exclusive strata based on a characteristic (stratification variable), then perform independent simple random sampling within each stratum.

4.2 Procedure

Step 1: Choose a stratification variable
        ↓
Step 2: Divide the population into L mutually exclusive strata (H₁, H₂, ..., Hₗ)
        ↓
Step 3: Perform simple random sampling independently within each stratum
        ↓
Step 4: Combine all stratum samples into the final sample

4.3 Criteria for Choosing Stratification Variables

• Must be highly correlated with the variable under study
• Individuals within each stratum should be homogeneous (similar)
• Individuals across different strata should be heterogeneous (different)

4.4 Practical Example

Case: Studying valuation levels in China's A-share market. Population = all A-share stocks.

• Stratification variable: Industry sector (financials, technology, consumer, healthcare, energy, etc.)
• Reason: P/E ratios differ dramatically across sectors (tech P/E >> financials P/E)
• Approach: Randomly select 30 stocks from each sector
• Effect: Ensures every sector is represented, avoiding bias such as "all technology stocks"

4.5 Two Allocation Methods

4.6 Advantages

4.7 Disadvantages

5. Systematic Sampling

5.1 Definition

After arranging the population in some order, select every k-th individual at a fixed interval. k is called the sampling interval.

5.2 Procedure

$$k = \frac{N}{n}$$

• Randomly select a starting point (between 1 and k)
• Select every k-th individual thereafter
• The i-th selected individual number = starting point + (i−1) × k

5.3 Practical Example

Case: A factory produces 10,000 units per day. A quality inspector needs to sample 200 units for testing.

• k = 10,000 / 200 = 50
• Randomly select a starting point between 1 and 50, say 23
• Select units #23, #73, #123, #173, ...

This is a classic application of interval sampling.

5.4 Advantages

5.5 Disadvantages

5.6 ⚠️ Periodicity Bias — High-Frequency CFA Topic

Classic example: An analyst samples a retailer's daily sales. Takes one data point per day:

• Sampling interval k = 7 days
• If the selected day always falls on Saturday → Saturday sales are abnormally high → severely overestimates average daily sales
• This is Periodicity Bias

Mitigation: Check whether the population exhibits a cyclical pattern; if so, switch to simple random sampling.

6. Comparison of the Three Probability Sampling Methods

7. Non-Probability Sampling (Overview)

CFA Level 1 requires knowing that non-probability sampling cannot be used for statistical inference because sampling error cannot be calculated.

8. Key Conceptual Distinctions

8.1 Sampling Error vs Sampling Bias

8.2 Sampling Frame

Definition: The "list" or "directory" from which the sample is drawn.

Key Risk: Sampling frame ≠ Population → Sampling Frame Bias

Example: Using a telephone directory as the sampling frame to survey household income → misses people without landlines (who may have lower incomes) → overestimates income

9. CFA Exam Decision Tree

When faced with a sampling method multiple-choice question, follow this logic:

【Diagnostic Flow】
  ├── Does the question describe "grouping first, then sampling"?
  │   ├── Yes → Stratified Random Sampling (key: random within strata)
  │   └── No → Continue
  ├── Does the question describe "every k-th individual"?
  │   ├── Yes → Systematic Sampling (⚠️ check for periodicity risk)
  │   └── No → Continue
  └── Does the question describe "completely random, no pre-processing"?
      └── Yes → Simple Random Sampling

10. Practice Questions

Question 1

An analyst divides 50 high schools in a city into "key schools" and "regular schools," then randomly selects 5 schools from each category for a survey. This sampling method is:

A. Simple random sampling B. Stratified random sampling C. Systematic sampling D. Judgmental sampling

Question 2

A supermarket wants to survey 500 out of its 20,000 members. After sorting by member ID, every 40th member is selected. This method is:

A. Simple random sampling B. Stratified random sampling C. Systematic sampling D. Quota sampling

Question 3 (Tricky)

Regarding stratified random sampling, which of the following statements is correct?

A. Stratified sampling is always more precise than simple random sampling B. Individuals within a stratum should be as different as possible C. The stratification variable must be highly correlated with the study variable to improve precision D. Every stratum must use the same sample size in stratified sampling

Question 4

Which of the following sampling methods must have a complete list of the population?

A. Simple random sampling B. Systematic sampling C. Convenience sampling D. Judgmental sampling

Question 5 (Scenario)

An analyst wants to study the dividend policies of all NYSE-listed companies. She sorts all stocks alphabetically by ticker and selects all companies whose tickers start with "A" as the sample. The most likely problem with this sample is:

A. Excessive sampling error B. Periodicity bias C. Sampling frame bias D. Systematic bias from convenience sampling

11. Answers and Explanations

【Question 1 Answer】B — Stratified Random Sampling

First classify into "key/regular" (stratification), then random-sample within each category → this is the standard procedure for stratified random sampling.

🧠 Keywords for identifying stratified sampling: "first divide into... groups/categories/strata" + "randomly select within each"

【Question 2 Answer】C — Systematic Sampling

Selecting at a fixed interval (every 40th member) → the defining characteristic of systematic sampling.

🧠 Keywords for systematic sampling: "every..." "fixed interval" "k-th". Note: sorting by member ID presents no periodicity risk since IDs are randomly assigned.

【Question 3 Answer】C

🧠 The efficiency of stratified sampling comes from: within-stratum homogeneity → low within-stratum variance → low overall standard error. If the stratification variable is unrelated to the study variable, this benefit does not materialize.

【Question 4 Answer】A — Simple Random Sampling

Simple random sampling requires giving every individual an equal probability of selection, which can only be ensured with a complete list.

• Systematic sampling: only requires knowing population size N and the ordering, not a one-to-one complete list
• Convenience / Judgmental sampling: non-probability methods, no list required

【Question 5 Answer】D — Systematic bias from convenience sampling

Selecting all companies whose tickers start with "A" → this is convenience sampling (easy to find), not random sampling.

• Alphabetical order may correlate with company characteristics (e.g., Chinese companies listed in the US often cluster around certain letters)
• This sample cannot represent the entire NYSE
• Severe selection bias exists

🧠 This question tests the limitation of non-probability sampling — it cannot be used for statistical inference. "Ticker starts with A" ≠ random.

12. Lesson Summary

📚 Next Lesson L126: Sampling Error and Standard Error — Where does sampling error come from? How is standard error calculated? How does the Central Limit Theorem guarantee that "the sample mean ≈ normal"?