📖 标准正态分布与 Z 分数

Quantitative Methods — Probability Module · Lesson 7

I. Lesson Positioning

L120 covered the normal distribution: a bell-shaped symmetric curve fully defined by μ and σ, with the 68-95-99.7 empirical rule for quick probability estimates. But that rule only gives you the three integer σ multiples. To precisely compute P(return > 2.3%), you need the standard normal distribution and z-scores.

II. Core Concepts

1. Why Standardize?

The normal distribution N(μ, σ²) has infinitely many possibilities — different μ, different σ, generating countless different bell curves. Publishing a separate probability table for every (μ, σ) combination is impossible.

Solution: transform all normal distributions onto a single "ruler." Just as you'd convert USD, EUR, and JPY into a single base currency to compare them — the z-score is that "universal conversion tool."

2. Standard Normal Distribution

$$\text{Denoted: } Z \sim N(0, 1)$$

$$f(z) = \frac{1}{\sqrt{2\pi}} \cdot e^{-\frac{z^2}{2}}$$

🧠 The standard normal has only one z-table — look it up once, use it everywhere.

3. Z-Score

The z-score answers: "How many standard deviations is X away from the mean?"

$$z = \frac{X - \mu}{\sigma}$$

Example 1: Standardization

X ~ N(100, 25), i.e., μ = 100, σ = 5:

📊 Standardization strips away the original units (%, dollars, points), leaving only unit-free σ multiples.

4. Four Z-Table Lookup Scenarios (Exam Core)

Abbreviated z-table (CFA exam common values):

Four scenarios:

Handling negative z — use symmetry:

$$P(Z \leq -z) = 1 - P(Z \leq z)$$

Example 2: Real Finance Problem

A fund's monthly return ~ N(0.8%, 2.5%²). Probability of a loss exceeding 2%:

$$z = \frac{-2\% - 0.8\%}{2.5\%} = -1.12$$

P(Z ≤ −1.12) = 1 − P(Z ≤ 1.12) = 1 − 0.8686 = 0.1314 ≈ 13.14%

📊 This fund has roughly a 13% chance of losing more than 2% in any given month.

5. Critical Z-Values — Must Memorize

🎯 Mnemonic: "One-tail 95 → 1.645, two-tail 95 → 1.96, one-tail 99 → 2.33, two-tail 99 → 2.58"

6. Reverse Standardization: From Probability Back to X

$$X = \mu + z \cdot \sigma$$

Example 3: VaR Calculation

Portfolio daily return ~ N(0.05%, 1.1%²). Daily VaR at 95% confidence:

• 95% confidence → P(Z ≤ z) = 0.05 → z = −1.645
• VaR = 0.05% + (−1.645)(1.1%) = −1.76%

📊 At 95% confidence, the max single-day loss is 1.76%.

7. Financial Applications of Z-Scores

III. Key Formulas

IV. Common Pitfalls

V. Practice Questions

【Questions】

Q1 (Concept) X ~ N(50, 25). When computing the z-score, the correct approach is:

A. Compute z = (X − 50) / 5, because σ = 5
B. Compute z = (X − 50) / 25, because σ² = 25
C. Compute z = X − 50 / 5, because subtraction takes priority over division
D. No z needed; look up X directly in the standard normal table

Q2 (Calculation) X ~ N(80, 144). The value of P(X > 95) is closest to:

A. 0.1056
B. 0.1250
C. 0.1587
D. 0.2112

Q3 (Calculation) X ~ N(200, 400). If P(X < k) = 0.8413, then k is closest to: (Given: P(Z ≤ 1.00) = 0.8413)

A. 210
B. 220
C. 240
D. 260

Q4 (Comprehensive) Standardized returns (z-scores) for three stocks:

Rank by z-score from highest to lowest (best → worst performer):

A. C > A > B
B. A > C > B
C. C > B > A
D. A > B > C

Q5 (Advanced Concept) Which statement about z-scores is incorrect?

A. A z-score measures the number of standard deviations from the mean
B. Z-scores can be computed even when the underlying distribution is non-normal
C. z > 0 means X is below the mean
D. After standardization, any dataset's sample mean becomes 0 and sample standard deviation becomes 1

【Answers & Explanations】

A1: A

• A ✅ σ² = 25 ⇒ σ = 5, so z = (X − 50)/5 is correct
• B ❌ The denominator must be standard deviation σ = 5, not variance σ² = 25
• C ❌ Subtraction first, then division — need parentheses: (X − 50)/5
• D ❌ Raw X values cannot be used directly with the standard normal table — must standardize first

A2: A — 0.1056

Calculation: - σ² = 144 ⇒ σ = 12 - z = (95 − 80) / 12 = 15/12 = 1.25 - P(Z ≤ 1.25) = 0.8944 - P(Z > 1.25) = 1 − 0.8944 = 0.1056

⚠️ Note: Compute z accurately first, then distinguish between left-tail and right-tail conversions.

A3: B — 220

• σ² = 400 ⇒ σ = 20
• P(Z ≤ 1.00) = 0.8413 ⇒ z = 1.00
• k = μ + z·σ = 200 + 1.00 × 20 = 220

A4: A — C > A > B

Compute each z-score: - Stock A: z_A = (5.2 − 3.0)/1.5 = 2.2/1.5 = 1.467 - Stock B: z_B = (−1.8 − 0.5)/0.8 = −2.3/0.8 = −2.875 - Stock C: z_C = (8.0 − 4.0)/2.0 = 4.0/2.0 = 2.000

Ranking: C (2.00) > A (1.47) > B (−2.88)

📊 Z-scores not only rank performance but quantify "how much better": C is 2σ above its mean, A is 1.47σ above, B is 2.88σ below.

A5: C

• A ✅ Definition of z-score: number of standard deviations from the mean
• B ✅ Z-score is purely a mathematical transformation: (X − μ)/σ, no normality assumption required
• C ❌ z > 0 means X is above the mean (numerator X − μ > 0)
• D ✅ After standardization, the sample mean always becomes 0 and sample standard deviation becomes 1 (mathematical property)

D may sound "too absolute," but it's a mathematical certainty: subtracting the mean makes the new mean 0, and dividing by σ makes the new standard deviation 1. This holds regardless of distribution shape.

📚 Next Lesson L122: Log-normal Distribution — why asset prices use log-normal instead of normal modeling