---
id: wiki-2026-0508-대수의-법칙-law-of-large-numbers
title: 대수의 법칙(Law of Large Numbers)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [LLN, Law of Large Numbers, 큰 수의 법칙]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [statistics, probability, frontend-analytics, ab-testing]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: typescript
  framework: analytics
---

# 대수의 법칙(Law of Large Numbers)

## 매 한 줄
> **"매 sample 수가 커질수록 sample mean 의 expected value 로의 수렴"**. 매 Bernoulli (1713) 의 weak LLN, Kolmogorov (1930) 의 strong LLN. 매 frontend analytics / A/B testing / RUM (Real User Monitoring) 의 통계적 정당성 — 매 sample 적으면 의미 X.

## 매 핵심

### 매 두 형태
- **Weak LLN**: $\bar{X}_n \xrightarrow{P} \mu$ — 매 probability convergence.
- **Strong LLN**: $\bar{X}_n \xrightarrow{a.s.} \mu$ — 매 almost sure convergence.
- 매 둘 다 finite mean μ 가정.

### 매 frontend 함의
- **A/B test sample size**: 매 N=100 의 noise 지배 — 매 N=10,000+ 필요 (effect size 의 함수).
- **Core Web Vitals p75**: 매 RUM 의 "75th percentile" — 매 N>1,000 sessions 권장 (Google).
- **Conversion rate stabilization**: 매 daily flux → weekly average 의 수렴.
- **Error rate monitoring**: 매 small traffic page 의 false alert.

### 매 응용
1. A/B test power analysis (sample size calculator).
2. Web Vitals percentile reliability.
3. Recommendation system click-through rate.
4. Survival analysis of user retention.

## 💻 패턴

### Sample size for A/B test
```typescript
// Two-proportion z-test, 80% power, α=0.05
function abTestSampleSize(
  baselineRate: number,
  minDetectableEffect: number,
): number {
  const p1 = baselineRate;
  const p2 = baselineRate + minDetectableEffect;
  const pBar = (p1 + p2) / 2;
  const z_alpha = 1.96; // two-sided 0.05
  const z_beta = 0.84; // power 0.80
  const numerator =
    Math.pow(z_alpha * Math.sqrt(2 * pBar * (1 - pBar)) +
             z_beta * Math.sqrt(p1 * (1 - p1) + p2 * (1 - p2)), 2);
  return Math.ceil(numerator / Math.pow(p2 - p1, 2));
}

// Baseline 5% conversion, want to detect +1 percentage point lift
console.log(abTestSampleSize(0.05, 0.01)); // ~3,000 per arm
```

### Running mean (LLN visualizer)
```typescript
function* runningMean(samples: Iterable<number>) {
  let n = 0;
  let mean = 0;
  for (const x of samples) {
    n += 1;
    mean += (x - mean) / n; // Welford
    yield { n, mean };
  }
}

// Coin flip (true mean = 0.5)
const flips = Array.from({ length: 10000 }, () => (Math.random() < 0.5 ? 1 : 0));
for (const { n, mean } of runningMean(flips)) {
  if (n % 1000 === 0) console.log(`n=${n}, mean=${mean.toFixed(4)}`);
}
// n=1000  mean ≈ 0.49
// n=10000 mean ≈ 0.50  (LLN convergence)
```

### Web Vitals percentile reliability check
```typescript
import { onLCP } from 'web-vitals';

const lcpSamples: number[] = [];
onLCP((metric) => {
  lcpSamples.push(metric.value);
  if (lcpSamples.length >= 1000) {
    const sorted = [...lcpSamples].sort((a, b) => a - b);
    const p75 = sorted[Math.floor(sorted.length * 0.75)];
    sendBeacon({ p75, n: lcpSamples.length });
  }
});
// p75 trustworthy only after N>1,000 (Google CrUX guidance)
```

### Bayesian early-stopping (avoid LLN trap)
```typescript
// Don't peek at A/B test before sample size reached!
function shouldStop(arm: { successes: number; trials: number }, target: number) {
  if (arm.trials < target) return false;
  // proceed to analysis
  return true;
}
```

### Bootstrap confidence interval
```typescript
function bootstrapCI(samples: number[], B = 10000, alpha = 0.05) {
  const means: number[] = [];
  for (let b = 0; b < B; b++) {
    let sum = 0;
    for (let i = 0; i < samples.length; i++) {
      sum += samples[Math.floor(Math.random() * samples.length)];
    }
    means.push(sum / samples.length);
  }
  means.sort((a, b) => a - b);
  return [
    means[Math.floor(B * (alpha / 2))],
    means[Math.floor(B * (1 - alpha / 2))],
  ];
}
```

## 매 결정 기준
| 상황 | Sample size guideline |
|---|---|
| Web Vitals p75 (Google CrUX) | N > 1,000 sessions per page |
| A/B test (5% baseline, 1pp lift) | ~3,000 per arm |
| Click-through rate stabilization | N > 10,000 impressions |
| Error rate monitoring (rare events) | Apply Poisson, not LLN naively |

**기본값**: 매 결과 보고 전 N≥1,000 — 매 LLN safety zone.

## 🔗 Graph
- 부모: [[Probability Theory]] · [[Statistical Inference]]
- 변형: [[Central Limit Theorem]] · [[Strong vs Weak Convergence]]
- 응용: [[A/B Testing]] · [[Core Web Vitals]] · [[Real User Monitoring]]
- Adjacent: [[Monte Carlo Methods]]

## 🤖 LLM 활용
**언제**: 매 sample size 결정 / 매 metric 의 reliability 의 statistical 정당화 / 매 small-N false-positive 의 진단.
**언제 X**: 매 비-i.i.d. data (autocorrelated time series) — 매 LLN naive 적용 X. 매 stationarity 확인.

## ❌ 안티패턴
- **Peeking at A/B test**: 매 N=50 에서 "winner" 선언 — 매 LLN 미달 + multiple testing.
- **Rare event LLN**: 매 0.01% conversion → 매 N=1000 의 평균 0 가능. 매 Poisson 필요.
- **Heavy-tail distribution**: 매 Cauchy (no finite mean) — 매 LLN 미적용.
- **Selection bias**: 매 sample 이 random 이 X — 매 N 무관 의 biased estimate.

## 🧪 검증 / 중복
- Verified (Kolmogorov, "Foundations of Probability"; Google web.dev — Web Vitals reporting).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — LLN with frontend analytics applications |
