---
id: wiki-2026-0508-standard-deviation-and-variance
title: Standard Deviation and Variance
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [SD, Variance, Sigma]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [statistics, variance, dispersion, foundations]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: numpy-scipy
---

# Standard Deviation and Variance

## 매 한 줄
> **"매 spread = E[(X-μ)²] 의 sqrt"**. Variance 의 expected squared deviation, SD 의 unit-scale spread. 1894 Pearson 이 명명 — 매 모든 statistics 의 가장 fundamental dispersion measure.

## 매 핵심

### 매 정의
- **Variance**: σ² = E[(X − μ)²] = E[X²] − (E[X])².
- **Standard deviation**: σ = √Var(X) — 매 same unit as X.
- **Sample variance**: s² = Σ(xᵢ − x̄)² / (n − 1) — 매 Bessel correction (n−1) 의 unbiased estimator.
- **Population variance**: σ² = Σ(xᵢ − μ)² / N.

### 매 property
- **Var(aX + b) = a²·Var(X)** — 매 shift invariant, scale 의 square.
- **Var(X + Y) = Var(X) + Var(Y) + 2·Cov(X, Y)** — independent 면 cov=0.
- **SD 의 robust X**: 매 outlier 의 영향 의 큼 — robust alt: MAD (Median Absolute Deviation), IQR.
- **Chebyshev**: P(|X−μ| ≥ kσ) ≤ 1/k² — 매 distribution-free.

### 매 응용
1. Risk (finance) — volatility = SD of returns.
2. Quality control — Six Sigma (process σ).
3. Z-score normalization — (x − μ) / σ.
4. Confidence interval — μ ± z·(σ/√n).

## 💻 패턴

### 1. NumPy — sample vs population
```python
import numpy as np
x = np.array([2, 4, 4, 4, 5, 5, 7, 9])
np.var(x)            # 매 population (ddof=0) → 4.0
np.var(x, ddof=1)    # 매 sample (Bessel) → 4.571
np.std(x, ddof=1)    # 매 2.138
```

### 2. Welford's online algorithm
```python
# 매 streaming, numerically stable, single-pass
def welford_update(state, x):
    n, mean, M2 = state
    n += 1
    delta = x - mean
    mean += delta / n
    M2 += delta * (x - mean)
    return n, mean, M2

n, mean, M2 = 0, 0.0, 0.0
for x in stream:
    n, mean, M2 = welford_update((n, mean, M2), x)
var = M2 / (n - 1)
```

### 3. Z-score normalization
```python
from sklearn.preprocessing import StandardScaler
scaler = StandardScaler()
X_scaled = scaler.fit_transform(X)  # 매 column-wise (x − μ) / σ
```

### 4. Rolling SD (pandas)
```python
import pandas as pd
returns = pd.Series(prices).pct_change()
volatility_30d = returns.rolling(30).std() * np.sqrt(252)  # annualized
```

### 5. Pooled variance (two groups)
```python
def pooled_var(x1, x2):
    n1, n2 = len(x1), len(x2)
    s1, s2 = np.var(x1, ddof=1), np.var(x2, ddof=1)
    return ((n1-1)*s1 + (n2-1)*s2) / (n1 + n2 - 2)
```

### 6. Bias-variance decomposition
```python
# E[(y - ŷ)²] = Bias² + Variance + σ²_noise
# 매 ML model selection 의 핵심
```

### 7. Robust alternatives — MAD
```python
from scipy.stats import median_abs_deviation
mad = median_abs_deviation(x, scale="normal")  # ~1.4826 * raw MAD ≈ σ for Gaussian
```

### 8. Two-pass safe variance (numerically)
```python
# Naive E[X²] - E[X]² → 매 catastrophic cancellation 의 위험 (large mean, small var)
# 매 2-pass: μ first, then Σ(x-μ)²
mean = x.sum() / n
var = ((x - mean) ** 2).sum() / (n - 1)
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Sample (inference) | ddof=1 (n−1 division) |
| Population (descriptive) | ddof=0 |
| Streaming / online | Welford |
| Outlier-heavy data | MAD, IQR (not SD) |
| Heavy-tail distribution | Quantile-based (P95/P5) |
| Volatility (finance) | Rolling SD × √(periods/year) |

**기본값**: `np.std(x, ddof=1)` — 매 sample SD, 매 unbiased point estimator.

## 🔗 Graph
- 부모: [[Statistics]] · [[Probability Theory]]
- 변형: [[Mutual-Information]] · [[Information-Entropy]]
- 응용: [[Regression-Analysis-Foundations]] · [[Statistical-Power]]
- Adjacent: [[Sampling-Techniques]] · [[Multivariate-Analysis]]

## 🤖 LLM 활용
**언제**: explanation of spread to non-technical, choosing appropriate measure given distribution.
**언제 X**: 매 numerical computation — `numpy` 의 사용.

## ❌ 안티패턴
- **ddof confusion**: NumPy default `ddof=0` (population) — 매 inference 시 `ddof=1` 의 명시.
- **SD on non-numeric / ordinal**: 매 ordinal scale 의 SD 의 의미 X.
- **Reporting SD without n**: 매 SE = σ/√n 의 더 informative.
- **Catastrophic cancellation**: naive Σx² − (Σx)²/n → use Welford or 2-pass.
- **SD assumes Gaussian-like**: 매 power-law 의 SD 의 unstable, 매 quantile 의 사용.

## 🧪 검증 / 중복
- Verified (Knuth TAOCP vol 2, Welford 1962, NIST/SEMATECH stats handbook).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — variance fundamentals, Welford, ddof, robust alt. |
