📖 离散程度：极差、MAD、方差、标准差

Topic: Don't Be Fooled by the Mean — Understanding Dispersion Is the Real Skill

1. Introduction: The Tale of Two Restaurants

Two restaurants with the same average customer rating of 4.0:

• Restaurant A: Ratings [4, 4, 4, 4, 4] — every customer agrees
• Restaurant B: Ratings [1, 1, 4, 5, 9] — some love it, some hate it

Same mean. Entirely different businesses.

Restaurant A is stable and predictable. Restaurant B is a roller coaster.

🔥 Core insight: The mean tells you "where the center is"; dispersion tells you "how spread out the data are." You must use both. Never one without the other.

2. What Is Dispersion?

2.1 Definition

Dispersion (Variability) measures how spread out or clustered data points are around their central tendency.

• Low dispersion → data tightly clustered around the mean (low volatility)
• High dispersion → data widely scattered (high volatility)

2.2 Relevance to Investing

📌 Dispersion is the mathematical foundation of risk measurement in the CFA curriculum. From variance to beta to VaR — everything starts with these four tools.

3. The Four Dispersion Measures at a Glance

4. Detailed Explanations

4.1 Range

Formula: Range = X_max − X_min

Advantage: Most intuitive — "the max minus the min."

Fatal flaw: Only uses two data points; ignores everything in between.

Practical Example:

Monthly returns for Fund A and Fund B: - A: [−2%, 1%, 2%, 3%, 5%] → Range = 5% − (−2%) = 7% - B: [−2%, −1%, 2%, 3%, 12%] → Range = 12% − (−2%) = 14%

Based on range alone, Fund B appears riskier. But Fund B has only one outlier (12%); all other months are similar to Fund A.

📌 The investment industry almost never uses range in isolation — a single outlier can completely distort it.

4.2 MAD — Mean Absolute Deviation

Formula:

$$\text{MAD} = \frac{\sum_{i=1}^{n} |X_i - \bar{X}|}{n}$$

Calculation steps: 1. Compute the mean X̄ 2. Compute the absolute deviation of each value from the mean: |Xᵢ − X̄| 3. Average all absolute deviations

Why absolute values?

Raw deviations (Xᵢ − X̄) always sum to zero (positives and negatives cancel out). Without absolute values, MAD would always be zero — meaningless.

Worked Example:

Dataset: [3, 5, 7, 9, 11]

X̄ = (3+5+7+9+11) / 5 = 7

MAD = (4+2+0+2+4) / 5 = 12/5 = 2.4

Interpretation: On average, each data point is 2.4 units away from the mean.

📌 MAD is easy to interpret and preserves original units, but absolute values are mathematically inconvenient (non-differentiable) — which is why variance and standard deviation dominate.

4.3 Variance — The Engine of Dispersion

4.3.1 Population Variance

Formula:

$$\sigma^2 = \frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N}$$

4.3.2 Sample Variance

Formula:

$$s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n-1}$$

4.3.3 Why Divide by n−1 Instead of n?

This is a must-know CFA Level 1 exam topic.

Intuition: The sample mean X̄ is computed from the sample itself, so it is "closer" to the sample data points than the true population mean μ would be. This makes deviations (Xᵢ − X̄) systematically smaller than (Xᵢ − μ).

Dividing by n−1 (instead of n) allocates the sum across fewer units — making the result larger and compensating for the downward bias.

Technical term: Bessel's Correction — makes the sample variance an unbiased estimator of the population variance.

🔑 Memory aid: "Population: N. Sample: n−1."

4.3.4 Variance Worked Example

Population data: [3, 5, 7, 9, 11], μ = 7

σ² = (16+4+0+4+16) / 5 = 40/5 = 8.0

4.3.5 The Core Flaw of Variance

Variance is expressed in squared units.

• If data are "returns in %" → variance unit is "%²" (percent squared?!)
• If data are "prices in dollars" → variance unit is "dollars²"

This is exactly why we need standard deviation — to bring us back to original units.

4.4 Standard Deviation — The "Final Form" of Dispersion

Formula:

Population standard deviation: $$\sigma = \sqrt{\sigma^2}$$

Sample standard deviation: $$s = \sqrt{s^2}$$

From the example above: σ = √8 = 2.828

Interpretation: Under a normal distribution, approximately 68% of observations fall within ±1 standard deviation of the mean.

5. Summary Comparison of the Four Measures

6. Chebyshev's Inequality

6.1 Definition

For any distribution, the proportion of observations falling within ± k standard deviations of the mean is at least:

$$1 - \frac{1}{k^2}, \quad k > 1$$

6.2 Application

📌 k=2 (≥75%) and k=3 (≥88.9%) are high-frequency CFA Level 1 exam points.

6.3 Comparison: Chebyshev vs. Normal Distribution

🔥 Chebyshev is the "floor" estimate — safe for any distribution. The normal distribution gives precise proportions but only applies when normality holds.

7. Coefficient of Variation (CV)

7.1 Why Do We Need CV?

Consider: Fund A has mean return 12%, standard deviation 8%. Fund B has mean return 5%, standard deviation 4%. Which carries more risk per unit of return?

Standalone σ cannot answer this — the two funds have different means, and standard deviations scale accordingly.

7.2 Formula

$$\text{CV} = \frac{s}{\bar{X}} \quad \text{or} \quad \frac{\sigma}{\mu}$$

7.3 Worked Example

• Fund A: CV_A = 8% / 12% = 0.667
• Fund B: CV_B = 4% / 5% = 0.800

Fund B has the higher CV → more volatility per unit of return.

7.4 Limitation of CV

⚠️ CV breaks down when the mean is close to zero or negative (dividing by a near-zero value produces absurd results).

8. Practical Application: Portfolio Risk Identification

Scenario

You are analyzing annualized daily volatility for three stocks (standard deviations already computed):

Analysis: - By standard deviation alone: Z is riskiest (σ=18%), Y is safest (σ=8%) - By CV: Y is riskiest (CV=1.00), X is most efficient (CV=0.80)

Conclusion: Y seems safe in absolute terms, but per unit of return, Y carries the highest risk — modest returns but disproportionately high volatility.

📌 The golden question in interviews and at work: "The volatility is high, sure — but relative to the return?" — that question leads straight to CV.

9. Summary: Dispersion Toolkit

🔥 Remember: No one makes decisions based on the mean alone. Dispersion is the language of risk. Learn it, and you will never be fooled by numbers.

10. Practice Questions

Question 1

Dataset: [2, 4, 4, 4, 5, 5, 7, 9], n = 8.

Calculate the range.

A. 5 B. 6 C. 7 D. 8

Question 2

Population data: [10, 12, 14, 16, 18]

Calculate the population variance σ².

A. 8 B. 10 C. 6 D. 7

Question 3

Using the same data, calculate the population standard deviation σ (to 2 decimal places).

A. 2.83 B. 3.16 C. 2.45 D. 3.00

Question 4

Regarding the use of n−1 as the denominator in sample variance, which of the following statements is most accurate?

A. It makes the variance smaller to be more conservative. B. One degree of freedom is lost, so a penalty is added. C. It makes the sample variance an unbiased estimator of the population variance. D. The population mean is unavailable, so the sample mean is used instead.

Question 5

According to Chebyshev's inequality, for any distribution, at least what proportion of observations falls within ±2.5 standard deviations of the mean?

A. 75% B. 84% C. 88.9% D. 80%

11. Answers and Explanations

Question 1 Answer: C. 7

Explanation: Range = X_max − X_min = 9 − 2 = 7

Question 2 Answer: A. 8

Explanation: μ = (10+12+14+16+18) / 5 = 70/5 = 14

σ² = (16+4+0+4+16) / 5 = 40/5 = 8

Question 3 Answer: A. 2.83

Explanation: σ = √8 ≈ 2.828 ≈ 2.83

Question 4 Answer: C

Explanation:

C ✅ Unbiased estimation is the precise mathematical purpose of Bessel's correction — E(s²) = σ². D is partially correct but incomplete — "no population mean" is the reason, but not the answer to "why n−1?" A is false — dividing by n−1 actually makes the variance larger, not smaller. B is close but imprecise — the degrees-of-freedom explanation has merit, but the CFA standard answer centers on unbiased estimation.

Question 5 Answer: B. 84%

Explanation: 1 − 1/k² = 1 − 1/(2.5)² = 1 − 1/6.25 = 1 − 0.16 = 0.84 = 84%

12. Key Formula Quick Reference

CFA Level 1 · L108 · Measures of Dispersion: Range, MAD, Variance, Standard Deviation · English Version · 2026-07-14