📖 分位数：四分位数、百分位数

Topic: Data Doesn't Speak — Quantiles Are Your Interpreter

1. Introduction: Why Every Fund Manager Claims "We Beat Half Our Peers"

You've seen this before: - "This fund outperforms 75% of peers" - "Our returns are in the top quartile"

Behind these statements lies a single mathematical tool: quantiles.

Quantiles are among the most practical and intuitive tools in statistics — they don't care about averages, only about your relative position in the group.

🔥 Core intuition: The mean tells you "how much everyone earns on average"; quantiles tell you "how much better you are than others." In investing, the latter often matters more.

2. What Is a Quantile?

2.1 Definition

A quantile is a value that divides a sorted dataset into equal-sized groups.

Given a sorted dataset, the p-th quantile is the value below which a proportion p of the data falls.

Critical prerequisite: data must be sorted (ascending) first. Unsorted quantiles are meaningless.

2.2 General Formula

For n data points, to find the y-th quantile:

Position index: L_y = (n + 1) × y

where y is the quantile point (for percentiles, y = k/100).

3. Three Core Quantile Types

3.1 Quartiles

Divide data into 4 equal parts.

Real-World Example: Fund Ratings

A platform ranks 1,000 funds by annualized return: - Q₁ = return of the 250th fund: laggard cutoff - Q₂ = return of the 500th fund: market median - Q₃ = return of the 750th fund: top-performer cutoff

If your fund exceeds Q₃ → you're in the top 25%. Time for the marketing brochure 📈

3.2 Percentiles

Divide data into 100 equal parts.

The k-th percentile = k% of data ≤ this value.

Key Relationships: - P₂₅ = Q₁ - P₅₀ = Q₂ = Median - P₇₅ = Q₃

Real-World Example: Exam Scores

The CFA exam passing score is not a fixed number — it is based on the Minimum Passing Score (MPS), which is essentially a calibrated percentile concept: - If your score exceeds the MPS-aligned percentile → you pass - It's a relative competition against your cohort, not an absolute score threshold

3.3 Quintiles and Deciles

4. Calculation Method (Key Exam Topic)

4.1 Position Formula

For n sorted data points, the position of the y-th quantile (0 < y < 1):

L_y = (n + 1) × y

4.2 Two-Step Procedure

1. Compute L_y = (n+1) × y
2. If L_y is an integer → directly take the L_y-th data value
3. If L_y is not an integer → linear interpolation

4.3 Linear Interpolation

Let L_y = a + d, where a is the integer part and d is the fractional part (0 < d < 1):

Quantile = X_a + d × (X_{a+1} − X_a)

4.4 Full Worked Example

Dataset (sorted): 3, 5, 7, 8, 9, 11, 13, 15

n = 8

Find Q₁ (25th percentile): - L = (8+1) × 0.25 = 9 × 0.25 = 2.25 - a = 2, d = 0.25 - Q₁ = X₂ + 0.25 × (X₃ − X₂) = 5 + 0.25 × (7−5) = 5 + 0.5 = 5.5

Find Q₂ (Median / 50th percentile): - L = (8+1) × 0.50 = 9 × 0.50 = 4.5 - a = 4, d = 0.5 - Q₂ = X₄ + 0.5 × (X₅ − X₄) = 8 + 0.5 × (9−8) = 8.5

Find Q₃ (75th percentile): - L = (8+1) × 0.75 = 9 × 0.75 = 6.75 - a = 6, d = 0.75 - Q₃ = X₆ + 0.75 × (X₇ − X₆) = 11 + 0.75 × (13−11) = 11 + 1.5 = 12.5

4.5 Interquartile Range (IQR)

IQR = Q₃ − Q₁

From the example: IQR = 12.5 − 5.5 = 7.0

📌 IQR is the core measure of dispersion for the middle 50% of data — it is unaffected by extreme values.

5. Three Key Applications of Quantiles

5.1 Box Plot — A Visualization Powerhouse

   Lower Fence    Q₁    Q₂    Q₃     Upper Fence
       |          |─────|─────|          |
   o---+----------+=====+=====+----------+---o
   Outlier              └── IQR ──┘       Outlier

Box Plot Construction Rules: - Box: Q₁ to Q₃ - Line inside box: Q₂ (median) - Lower Whisker: minimum value within Q₁ − 1.5 × IQR - Upper Whisker: maximum value within Q₃ + 1.5 × IQR - Points beyond whiskers: outliers

5.2 Outlier Detection

Mild outliers: more than 1.5 × IQR from Q₁ or Q₃ Extreme outliers: more than 3.0 × IQR from Q₁ or Q₃

📌 This is a common CFA Level 1 exam topic: IQR-based outlier detection vs. standard deviation-based detection.

5.3 Relative Position Assessment

Application: You want to know where a particular stock stands among its peers.

• Sort 500 peer stocks by return
• Your stock at the P₈₀ quantile → it beats 80% of peers
• This is far more compelling than saying "the average return is 15%."

6. Pitfalls and Common Traps

6.1 Trap 1: Forgetting to Sort

❌ Wrong: taking a value at some position from raw data ✅ Correct: sort first, then compute quantiles

6.2 Trap 2: Formula Confusion

Different textbooks may use slightly different coefficients (some use n, some use n−1). CFA Level 1 uniformly uses L_y = (n+1) × y.

6.3 Trap 3: Quantile Point vs. Quantile Value

• Quantile point (y): a proportion, e.g., 0.25, 0.75
• Quantile value: the data value at that quantile point
• Example: P₂₅ (25th percentile) = 5.5, where 0.25 is the quantile point, 5.5 is the quantile value

6.4 Trap 4: The Robustness of the Median

The median is unaffected by extreme values; the mean gets severely distorted.

The mean skyrockets from 8.875 to 25.75, while the median stays rock-solid. This is why income statistics often use the median to reveal the truth rather than the "average" that gets distorted by the ultra-rich.

7. Summary: Quantile Toolkit Cheat Sheet

🔥 Remember: statistics isn't about crunching numbers — it's about telling stories. Quantiles help you tell the story of "where I stand."

8. Practice Questions

Question 1

Dataset (sorted): [2, 4, 6, 8, 10, 12, 14, 16, 18, 20], n = 10.

Find Q₁.

A. 5.5 B. 5.25 C. 6.0 D. 5.75

Question 2

Using the same dataset, find the 80th percentile P₈₀.

A. 16.8 B. 16.0 C. 17.2 D. 17.8

Question 3

Which of the following statements about quantiles is incorrect?

A. The median is the 50th percentile and also the second quartile Q₂ B. Data must be sorted before computing quantiles C. IQR is highly sensitive to extreme values and therefore unsuitable for outlier detection D. P₂₅ and Q₁ are equivalent

9. Answers and Explanations

Question 1 Answer: D. 5.75

Explanation:

Q₁ → y = 0.25 L = (10+1) × 0.25 = 11 × 0.25 = 2.75 a = 2, d = 0.75 Q₁ = X₂ + 0.75 × (X₃ − X₂) = 4 + 0.75 × (6 − 4) = 4 + 1.5 = 5.75

Question 2 Answer: A. 16.8

Explanation:

P₈₀ → y = 0.80 L = (10+1) × 0.80 = 11 × 0.80 = 8.8 a = 8, d = 0.8 P₈₀ = X₈ + 0.8 × (X₉ − X₈) = 16 + 0.8 × (18−16) = 16 + 1.6 = 16.8

Question 3 Answer: C

Explanation:

C is incorrect. The IQR is specifically designed to be insensitive to extreme values (because it only considers Q₁ and Q₃ — the middle 50% of data). IQR-based outlier detection (the box plot method) is one of the standard tools in statistics for identifying outliers, precisely because of IQR's robustness to extreme values.

A ✅ Median = P₅₀ = Q₂ B ✅ Sorting is a prerequisite for quantile calculation C ❌ Gets it exactly backwards — IQR is insensitive to extremes, making it naturally suited for outlier detection D ✅ P₂₅ and Q₁ are both the 25% quantile point

10. Key Formula Quick Reference

CFA Level 1 · L107 · Quantiles: Quartiles and Percentiles · English Version · 2026-07-13