Table of Contents
What Is Normal Approximation?
Normal approximation uses the normal distribution to estimate probabilities from discrete distributions like the binomial or Poisson. As sample sizes increase, these distributions approach the normal distribution shape (Central Limit Theorem).
This technique was essential before computers made exact calculations easy. It remains useful for understanding concepts, quick mental calculations, and situations where exact computation is impractical.
Formulas
The ±0.5 is the continuity correction, which adjusts for approximating a discrete distribution with a continuous one.
Rules of Thumb
| Distribution | Condition for Approximation |
|---|---|
| Binomial | np ≥ 5 and n(1-p) ≥ 5 |
| Poisson | λ ≥ 10 (some say ≥ 5) |
Frequently Asked Questions
What is the continuity correction?
Since we are approximating a discrete distribution with a continuous one, we add or subtract 0.5 to improve accuracy. For P(X ≤ k), use P(Z ≤ (k + 0.5 - μ)/σ). This accounts for the width of the bars in a histogram.
How accurate is the normal approximation?
Accuracy improves with larger n (for binomial) or larger λ (for Poisson). For binomial, it works well when np and n(1-p) are both at least 10. The approximation is poorest near p = 0 or p = 1.
Why not just use exact calculations?
With modern computers, exact calculations are preferred. However, the normal approximation remains valuable for understanding the CLT, for very large n where exact binomial coefficients overflow, and for quick mental estimates.