📖 集中趋势：均值、中位数、众数

Topic: The "Anchors" of Data — Three Types of Means and Their Rightful Place

I. Introduction: Three Questions for an Investment Analyst

Imagine you're faced with three datasets, each representing a key investment metric:

Three scenarios, three answers — corresponding to arithmetic mean, geometric mean, and weighted mean.

If you use only one type of "average" for every situation, you may end up buying "fake cheapness" or "exaggerated returns."

The goal of this lesson: thoroughly understand which measure of central tendency to use when, and when absolutely not to use it.

II. Core Concept: Where Is the Center?

2.1 Definition of Central Tendency

Central tendency refers to the value around which a dataset clusters. It is the first step in descriptive statistics — before any analysis, ask: what is the "typical value" of the data?

Three classic measures: - Mean — the arithmetic average, influenced by every data point - Median — the middle value after sorting, unaffected by extreme values - Mode — the most frequently occurring value, usable for qualitative data

📌 CFA Level I Focus: know which measure of central tendency can be used for each measurement scale.

III. Deep Dive: The Three Types of Means

3.1 Arithmetic Mean — The "Default" Average

Formula:

$$\bar{x} = \frac{x_1 + x_2 + \cdots + x_n}{n} = \frac{\sum_{i=1}^{n} x_i}{n}$$

Characteristics: - The most intuitive and commonly used mean - Every data point carries equal weight - Fatal flaw: extremely sensitive to outliers

Real-World Example: You analyze the performance of 10 fund managers. Nine have annual returns of 5%-12%, but the tenth blew up at -60%. The arithmetic mean may be dragged down to 2%, yet 90% of managers performed decently.

When to Use: - Normally distributed data without severe skewness - When computing variance and standard deviation (variance is based on arithmetic mean) - Forecasting expected values for the next period

When NOT to Use: - Data with extreme outliers (the mean gets distorted) → Use median - Measuring growth rates over time → Use geometric mean

3.2 Geometric Mean — "The Friend of Compound Returns"

Formula:

$$G = \sqrt[n]{x_1 \times x_2 \times \cdots \times x_n} = (x_1 \times x_2 \times \cdots \times x_n)^{1/n}$$

For return data, each $x_i = (1 + r_i)$:

$$G = \sqrt[n]{(1+r_1)(1+r_2)\cdots(1+r_n)} - 1$$

Characteristics: - Reflects the true speed of compound growth - Always ≤ Arithmetic Mean (unless all values are equal) - Sensitive to negative or zero values (geometric mean may become meaningless)

Real-World Example — Why funds advertise "arithmetic mean" instead of "geometric mean":

A fund's three-year returns: - Year 1: +50% - Year 2: -30% - Year 3: +50%

Arithmetic Mean: (50% + (-30%) + 50%) / 3 = 23.3% → Looks very attractive 😍

Geometric Mean: ³√(1.5 × 0.7 × 1.5) - 1 = ³√1.575 - 1 ≈ 16.3% → This is the real growth rate 😑

Verification: Invest $100: - End of Year 1: 100 × 1.5 = 150 - End of Year 2: 150 × 0.7 = 105 - End of Year 3: 105 × 1.5 = 157.5

Using arithmetic mean 23.3% compounded: 100 × 1.233³ = 187.5 (overstated by $30!)

🔑 CFA Must-Know: For time-series returns, use geometric mean for compound growth rates. For cross-sectional data, use arithmetic mean for central location.

Real-World Pitfalls: - Fund marketing materials showing "average annual return" as arithmetic mean → likely inflated - When reviewing performance, look for CAGR (Compound Annual Growth Rate) → that's geometric mean - The greater the volatility, the larger the gap between arithmetic and geometric mean

3.3 Harmonic Mean — "The DCA (Dollar-Cost Averaging) Ally"

Formula:

$$H = \frac{n}{\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}} = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$

Characteristics: - Always ≤ Geometric Mean ≤ Arithmetic Mean - Highly sensitive to very small values - Primarily used in "fixed-dollar-amount purchasing" scenarios

Real-World Example — Harmonic Mean in Dollar-Cost Averaging:

You invest $1,000 per month into an ETF: - Month 1: price $20/share → buy 50 shares - Month 2: price $25/share → buy 40 shares - Month 3: price $10/share → buy 100 shares

Total investment: $3,000; Total shares: 190

Harmonic Mean Price: 3 / (1/20 + 1/25 + 1/10) = 3 / (0.05 + 0.04 + 0.10) = 3 / 0.19 = $15.79

Arithmetic Mean Price: (20 + 25 + 10) / 3 = $18.33

Actual cost per share = 3000 / 190 = $15.79 ← Exactly the harmonic mean!

🔑 The harmonic mean emerges naturally when you buy with a fixed dollar amount rather than a fixed number of shares.

CFA Level I Harmonic Mean Focus Areas: 1. Use harmonic mean for calculating "average cost per share" 2. Harmonic ≤ Geometric ≤ Arithmetic (unless all values equal) 3. Harmonic mean is highly sensitive to extremely small values

3.4 Weighted Mean — "The Truth About Index-Level P/E"

Formula:

$$\bar{x}_w = \frac{w_1 x_1 + w_2 x_2 + \cdots + w_n x_n}{w_1 + w_2 + \cdots + w_n} = \frac{\sum w_i x_i}{\sum w_i}$$

Characteristics: - Each data point is assigned a different level of importance (weight) - Simplest case: equal weights → reduces to arithmetic mean

Real-World Example — Why index P/E cannot use simple arithmetic mean:

In the S&P 500: - Apple: P/E = 30, market cap $3 trillion - A small-cap stock: P/E = 50, market cap $20 billion

Simple arithmetic mean: (30 + 50) / 2 = 40 ✗ — Misleading!

Weighted mean (by market cap): (30 × 3000 + 50 × 20) / (3000 + 20) ≈ 30.13 ✓

Correct index P/E is always market-cap weighted!!

📌 CFA Focus: Index calculations, portfolio returns, WACC — all rely on the concept of weighted averages.

3.5 Summary: Three Means Relationship

Relationship: Harmonic ≤ Geometric ≤ Arithmetic

Equality holds only when: all values are identical.

IV. Median — The Robust Measure That Ignores Outliers

4.1 Definition and Calculation

Median: the middle value after sorting data from smallest to largest.

• n is odd: the (n+1)/2 th value
• n is even: arithmetic mean of the two middle values

4.2 Median vs. Mean — When to "Betray" the Mean

Classic Example: Average income vs. median income

Consider the monthly salaries (in $ thousands) of a 10-person company:

1.2, 1.3, 1.5, 1.5, 1.6, 1.7, 1.8, 2.0, 2.2, 35.0 (CEO)

• Arithmetic Mean: 49.8 / 10 = $49,800
• Median: average of 5th and 6th = (1.6 + 1.7) / 2 = $16,500

Which better represents a "typical employee's" salary? Clearly the median at $16,500.

When data is positively skewed (right-skewed): Mean > Median > Mode When data is negatively skewed (left-skewed): Mean < Median < Mode

4.3 Advantages of the Median

• ✅ Unaffected by extreme values (robustness)
• ✅ Suitable for ordinal data (e.g., Morningstar ratings)
• ✅ Suitable for skewed distributions
• ❌ Difficult to use in mathematical derivations (cannot directly derive variance from median)
• ❌ May be unstable with small samples

V. Mode — "The Most Popular Choice"

5.1 Definition

Mode: the value that appears most frequently in a dataset.

• Can have 0 modes (all values appear equally often)
• Can have 1 mode (Unimodal)
• Can have multiple modes (Bimodal / Multimodal)

5.2 The Unique Position of the Mode

The only measure of central tendency that can be used with nominal data!

Examples: - The most popular ETF category among investors → Mode is "Diversified Emerging Mkts" - The most commonly assigned fund rating → Mode is 4 stars

5.3 Limitations of the Mode

• Can be highly sensitive to small data changes (in a bimodal distribution, one extra data point may shift the mode)
• Does not reflect the overall distribution
• Limited meaning for continuous data (use modal class after grouping)

VI. Quantiles — A Finer Lens Than the Median

6.1 Common Quantiles

Position formula: $L_y = (n+1) \times \frac{y}{100}$

• If L is an integer → take the L-th value directly
• If L is not an integer → use linear interpolation

6.2 Real-World Example — Using Quantiles to Assess Fund Performance

A fund manager tells you: "My fund's return is in the top quartile."

How to verify? — His fund return > Q3 (75th percentile), meaning it beat 75% of peers.

• Q1 → worst 25%
• Q2 (Median) → right in the middle
• Q3 → best 25% → This is what "top quartile" means

VII. Comprehensive Comparison: When to Use What?

VIII. Practice Questions

Multiple Choice

Q1: A fund's returns over the past 4 years are +10%, -5%, +20%, +15%. To calculate the fund's compound annual growth rate (CAGR), which mean should be used? - A. Arithmetic Mean - B. Geometric Mean - C. Harmonic Mean - D. Weighted Mean

Q2: For which type of data can the median, but NOT the mean, be used as a measure of central tendency? - A. Normally distributed returns - B. Severely right-skewed income data - C. Z-scores from a standard normal distribution - D. Daily closing prices of an ETF

Q3: The relationship among harmonic, geometric, and arithmetic means is: - A. Arithmetic ≥ Geometric ≥ Harmonic (unless all values are equal) - B. Harmonic ≥ Geometric ≥ Arithmetic (unless all values are equal) - C. Geometric ≥ Arithmetic ≥ Harmonic (unless all values are equal) - D. There is no fixed relationship

Q4: Nominal data (e.g., investment style labels: Growth/Value/Blend) can only use which measure of central tendency? - A. Arithmetic Mean - B. Median - C. Mode - D. Geometric Mean

Q5: An analyst wants to calculate the average P/E ratio of S&P 500 constituents. Which method is most correct? - A. Simple arithmetic average of all 500 P/E ratios - B. Market-cap weighted average - C. Free-float weighted average - D. Median P/E

Q6: In a portfolio, if Q1 return = 2% and Q3 return = 12%, which of the following is correct? - A. 25% of periods had returns exceeding 12% - B. 50% of periods had returns between 2% and 12% - C. The median return must equal 7% - D. The minimum must be less than 2%

Q7: In a positively skewed (right-skewed) distribution, the relationship among mean, median, and mode is: - A. Mean < Median < Mode - B. Mean > Median > Mode - C. Mean = Median = Mode - D. Median > Mean > Mode

Answers

Q1: B — Time-series returns measure compound growth and must use geometric mean. CAGR = [(1.10)(0.95)(1.20)(1.15)]^(1/4) - 1 ≈ 9.47%. Arithmetic mean = (10% - 5% + 20% + 15%) / 4 = 10%, which overstates the true compound growth rate.

Q2: B — Income data is typically severely right-skewed (a few extremely high incomes pull the mean upward). The median better represents the "typical" income level. For normally distributed data, the mean and median are equal.

Q3: A — Arithmetic ≥ Geometric ≥ Harmonic, with equality only when all values are identical. This is a classic CFA Level I concept.

Q4: C — Nominal data (pure categorical labels) has no numerical values and no ordering. Only the mode can count which category appears most often. You cannot "average" Growth and Value.

Q5: B — Market-cap weighted P/E is the industry standard for index valuation. Simple arithmetic average would treat small-cap and large-cap stocks equally, distorting the index's true valuation level.

Q6: B — The interquartile range (Q1 to Q3) contains the middle 50% of the data (from the 25th to the 75th percentile), so 50% of periods had returns between 2% and 12%. A is incorrect — 25% exceed Q3. C is not necessarily true. D is not guaranteed.

Q7: B — In a positively skewed distribution, the long tail is on the right, pulling the mean upward. The mode sits at the peak (highest point), the median in the middle, and the mean furthest right: Mean > Median > Mode.

IX. Key Exam Points

X. Quick Memory Card

📌 One-Line Summary:

Arithmetic Mean → Cross-sectional expected value (most common)
Geometric Mean  → Compound growth rate / CAGR (mandatory for time series)
Harmonic Mean   → Dollar-cost averaging cost (equal-dollar purchases)
Median          → "Typical value" for skewed data
Mode            → "Most popular" for categorical data
Weighted Mean   → Market-cap weighting for indices/portfolios
Quantiles       → Relative rank (how many peers you beat)

CFA Level I · Quantitative Methods · Module 2.3 Begins | L105 Central Tendency · 2026-07-11