LESSON 5.3 · ABOUT 20 MINUTES
Central Limit Theorem
Approximate sums and averages when exact calculations are difficult.
By the end of this lesson
- Standardize a sum or average correctly.
- Recognize approximation conditions and continuity correction.
The approximation
For i.i.d. variables with finite mean μ and finite positive variance σ², a sufficiently large sum is approximately normal with mean nμ and variance nσ². The average has mean μ and variance σ²/n.
Sample size is not a guarantee
There is no universal sample size that makes every tail approximation accurate. Skewness, heavy tails, and how far into the tail you look matter. The finite-variance and independence assumptions are essential.
Discrete boundaries
For integer-valued sums with unit spacing, use a continuity correction: P(S≤k)≈P(N≤k+0.5). For P(S≥k), use k−0.5. Standardize only after adjusting the boundary.
Z = (Sₙ − nμ)/(σ√n); SD(X̄)=σ/√n
Worked example
100 independent claim amounts each have mean 10 and SD 4. Approximate P(total≤1080).
The total has mean 1000 and SD √100×4=40. z=(1080−1000)/40=2. Thus the probability is approximately Φ(2)=0.9772. No integer continuity correction is implied for continuous claim amounts.