📖 中心极限定理

Quantitative Methods — Sampling and Estimation · Lesson 2

1. Starting with a Question

Suppose you want to estimate the average P/E ratio of all stocks listed on the Shenzhen Stock Exchange. There are 2,800+ stocks in the population. You randomly select 100 stocks and compute a sample mean $\bar{x}$ = 15.3.

Question: How far is this 15.3 from the true population mean μ? If you repeated this sampling 1,000 times (each time drawing 100 stocks), what distribution would those $\bar{x}$ values follow?

The answer to this question is the Central Limit Theorem (CLT) .

2. Central Limit Theorem: Core Statement

2.1 Definition

Central Limit Theorem (CLT): For a population with mean μ and variance σ², if we draw simple random samples of size n, then as the sample size n becomes sufficiently large, the sampling distribution of the sample mean $\bar{x}$ approaches a normal distribution with mean μ and variance σ²/n.

In mathematical notation:

$$\bar{x} \sim N\left(\mu, \frac{\sigma^2}{n}\right) \quad \text{(when n is sufficiently large)}$$

2.2 Three Core Conclusions of CLT

🔥 Conclusion #3 is the essence of CLT: The population can be uniform, exponential, or even highly skewed — as long as n is large enough, the distribution of the sample mean approximates normality. This is what makes CLT so powerful.

3. Intuitive Understanding: Dice Experiment

3.1 Distribution of a Single Die

Rolling one fair die, the distribution of outcome X is discrete uniform (1–6, each 1/6):

P(X)
  │
  │  █  █  █  █  █  █
  │  █  █  █  █  █  █
  └──┴──┴──┴──┴──┴──
    1  2  3  4  5  6    → Not normal at all!

• μ = 3.5, σ² ≈ 2.92

3.2 Rolling Two Dice, Taking the Mean

Rolling 2 dice and computing the mean $\bar{x}_2$:

P(mean)
  │       █
  │     █████
  │   █████████
  │ █████████████
  └────────────────
  1  2  3  4  5  6   → Starting to look triangular

3.3 Rolling 30 Dice, Taking the Mean

Rolling 30 dice and computing the mean $\bar{x}_{30}$:

P(mean)
  │        ╭──╮
  │      ╱      ╲
  │    ╱          ╲
  │  ╱              ╲
  └────────────────────
  2.5  3.0  3.5  4.0  4.5  → Approximately normal!

💡 The population distribution of a die roll is completely non-normal (discrete uniform), yet when n = 30, the distribution of the sample mean is already very close to normal.

4. Key Conditions of CLT

4.1 How Large is "Sufficiently Large"?

📌 CFA Level 1 exam standard answer: n ≥ 30 is considered "sufficiently large." This is the widely-accepted Rule of Thumb.

4.2 Prerequisites

5. Mathematical Foundation of CLT

5.1 Sampling Distribution

Definition: The distribution of sample means computed from repeated samples of the same size drawn from the same population. This is the "sampling distribution of $\bar{x}$."

5.2 Three Properties

$$\text{Sampling distribution of the sample mean:}$$

5.3 Meaning of Standard Error

Standard Error (SE) is the standard deviation of the sampling distribution of the sample mean. It measures how much $\bar{x}$ varies around μ.

$$\text{Precision} \propto \frac{1}{\sqrt{n}}$$

🔑 Key insight: To double precision (halve the standard error), you need to quadruple the sample size. The cost of precision increases quadratically.

6. Practical Examples

6.1 Fund Performance Analysis

Scenario: A hedge fund claims its monthly return has a mean of 1.2% with a standard deviation of 3.5%. A regulator randomly selects 64 months of the fund's return records for audit.

Question: What is the standard error of the 64-month sample mean?

Solution: $$SE(\bar{x}) = \frac{\sigma}{\sqrt{n}} = \frac{3.5\%}{\sqrt{64}} = \frac{3.5\%}{8} = 0.4375\%$$

Interpretation: If the actual 64-month average return falls outside [1.2% − 1.96×0.4375%, 1.2% + 1.96×0.4375%] = [0.34%, 2.06%], we can challenge the claimed μ = 1.2% at the 5% significance level (foreshadowing hypothesis testing).

6.2 Customer Satisfaction Survey

Scenario: A restaurant chain's customer satisfaction scores have a population standard deviation of 15 points. Headquarters wants to estimate the national average score with a standard error of no more than 2 points.

Question: What is the minimum sample size required?

Solution: $$SE = \frac{\sigma}{\sqrt{n}} \leq 2$$ $$\frac{15}{\sqrt{n}} \leq 2$$ $$\sqrt{n} \geq 7.5$$ $$n \geq 56.25$$

→ At least 57 surveys required

6.3 Skewed Population of Stock Returns

Scenario: The daily returns of ChiNext (创业板) stocks are highly positively skewed (with extreme limit-up days). However, if we draw n = 50 ChiNext stocks and compute the average daily return...

Conclusion: Even though the population is severely skewed, n = 50 ≥ 30. According to CLT, the sampling distribution of the 50-stock mean return will still be approximately normal. Analysts can confidently use normal distribution critical values for inference.

7. Common Misconceptions

Misconception 1: CLT says "the sample itself becomes normal"

❌ Wrong: "Draw a large sample, and the data within the sample will approximate a normal distribution."

✅ Correct: CLT states that the sampling distribution of the sample mean approaches normality — not that individual observations within a single sample become normal.

Data within a single sample always reflects the shape of the population distribution. If you draw 1,000 observations from a skewed population, those 1,000 data points remain skewed. But if you draw 1,000 separate samples (each with n = 30), those 1,000 sample means will be approximately normal.

Misconception 2: n ≥ 30 is a hard rule

❌ Wrong: "n = 29 is insufficient; n = 30 is perfectly fine."

✅ Correct: n = 30 is merely a rule of thumb. The actual requirement depends on the degree of skewness. A mildly skewed population may be fine with n = 20; an extremely skewed one (e.g., insurance claim amounts) may require n = 100+.

Misconception 3: The population must be normal to apply CLT

❌ Wrong: "CLT only applies to normally distributed populations."

✅ Correct: The entire point of CLT is that the population does NOT need to be normally distributed. This is precisely why it is considered one of the most important theorems in statistics.

8. CLT and Investment Practice

9. CLT's Position in the CFA Curriculum

Descriptive Statistics → Probability → Random Variable Distributions (Normal, t, Chi-square, F)
                                              ↓
                                    Sampling and Estimation ← This section
                                              ↓
                                    【Central Limit Theorem】← L126 (Foundation)
                                              ↓
                                    Standard Error (L127)
                                              ↓
                                    Interval Estimation (L128–L129)
                                              ↓
                                    Hypothesis Testing (Upcoming modules)

⚠️ CLT is the cornerstone of inferential statistics. If CLT did not hold, everything that follows — confidence intervals, hypothesis testing, t-tests, regression inference — would all collapse.

10. Practice Questions

Question 1

The core conclusion of the Central Limit Theorem is:

A. Data from any population follows a normal distribution
B. When sample size is sufficiently large, the sampling distribution of the sample mean is approximately normal
C. When sample size is sufficiently large, each observation in the sample approximates normality
D. Sample means from a normal population always follow a normal distribution, regardless of sample size

Question 2

A simple random sample of n = 25 is drawn from a population with mean 100 and standard deviation 20. The standard error of the sample mean is:

A. 20
B. 4
C. 0.8
D. 5

Question 3

An analyst's population data is severely right-skewed (e.g., annual personal income). He draws a sample of n = 35. According to CLT, which of the following holds?

A. The 35 observations in the sample will be approximately normally distributed
B. The sampling distribution of the sample mean will be approximately normal
C. The sample mean equals the population mean
D. The sample standard deviation equals σ/√35

Question 4 (Calculation)

A stock strategy has a daily return standard deviation of 1.8%. If 81 trading days of returns are sampled to compute the average, what is the standard deviation of the sampling distribution of this average?

A. 1.8%
B. 0.2%
C. 0.022%
D. 0.18%

Question 5 (Conceptual)

Which of the following statements about the Central Limit Theorem is incorrect?

A. CLT requires that the sample be a simple random sample
B. The more skewed the population distribution, the larger the sample size needed for CLT to work
C. CLT guarantees that the sample mean exactly equals the population mean
D. CLT is the theoretical foundation of modern statistical inference methods

11. Answers and Explanations

【Answer 1】B — When sample size is sufficiently large, the sampling distribution of the sample mean is approximately normal

【Answer 2】B — 4

$$SE(\bar{x}) = \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{25}} = \frac{20}{5} = 4$$

【Answer 3】B — The sampling distribution of the sample mean will be approximately normal

【Answer 4】B — 0.2%

$$SE = \frac{1.8\%}{\sqrt{81}} = \frac{1.8\%}{9} = 0.2\%$$

【Answer 5】C — CLT guarantees that the sample mean exactly equals the population mean

12. Summary

📚 Next Lesson L127: Standard Error — Standard Error vs Standard Deviation? When does the t-distribution enter the picture? Complete mapping of sampling distributions.