Normal Approximation Calculator
Free normal approximation calculator. Approximate binomial, Poisson, hypergeometric with normal. Exa
Why This Statistical Analysis Matters
Why: Statistical calculator for analysis.
How: Enter inputs and compute results.
Normal Approximation — When Discrete Meets Continuous
Approximate binomial, Poisson, and hypergeometric distributions with the normal curve. Compare exact vs approximate probabilities (with and without continuity correction) and see relative error.
Real-World Scenarios — Click to Load
Distribution Type
Probability Type
Inputs
Exact PMF vs Normal Curve Overlay
Exact vs Approximate Probability Comparison
Relative Error: No CC vs With CC
Calculation Breakdown
For educational and informational purposes only. Verify with a qualified professional.
Key Takeaways
- • Binomial X ~ Bin(n,p) ≈ N(np, np(1-p)) when np ≥ 5 and n(1-p) ≥ 5
- • Poisson X ~ Poi(λ) ≈ N(λ, λ) when λ ≥ 10 (or λ ≥ 5 for rough approximation)
- • Hypergeometric ≈ Normal with μ = nK/N, σ² = nK(N-K)(N-n)/(N²(N-1))
- • Continuity correction: P(X=k) uses P(k-0.5 ≤ X ≤ k+0.5); P(X≤k) uses k+0.5; P(X≥k) uses k-0.5
- • Always compare exact vs approximate and report relative error
Did You Know?
How It Works
1. Binomial → Normal
μ = np, σ = √(np(1-p)). Valid when np ≥ 5 and n(1-p) ≥ 5. Use z = (k ± 0.5 - μ)/σ with continuity correction.
2. Poisson → Normal
μ = λ, σ = √λ. Valid when λ ≥ 10. P(X≤k) ≈ Φ((k+0.5-λ)/√λ); P(X≥k) ≈ 1 - Φ((k-0.5-λ)/√λ).
3. Hypergeometric → Normal
μ = nK/N, σ² = nK(N-K)(N-n)/(N²(N-1)). Finite population correction reduces variance compared to binomial.
4. Continuity Correction Rules
P(X=k): use Φ((k+0.5-μ)/σ) − Φ((k-0.5-μ)/σ). P(X≤k): use Φ((k+0.5-μ)/σ). P(X≥k): use 1 − Φ((k-0.5-μ)/σ).
5. Relative Error
Relative error % = |approximate − exact| / exact × 100. Continuity correction typically reduces error.
Expert Tips
When to Use Normal Approx
Binomial: np ≥ 5 and n(1-p) ≥ 5. Poisson: λ ≥ 10. Check validity before trusting results.
Always Use CC
Continuity correction improves accuracy, especially for P(X=k) and small k. Without CC, errors can exceed 10%.
Exact vs Approx
When in doubt, compute exact. Use approximation for quick estimates or when n is very large.
Hypergeometric Caveat
Sampling without replacement — variance is smaller. Use when n/N is not negligible.
Comparison: Exact vs Approximate
| Method | P(X=k) | P(X≤k) | P(X≥k) |
|---|---|---|---|
| Exact | PMF | CDF | 1-CDF(k-1) |
| Normal (no CC) | PDF×1 (poor) | Φ((k-μ)/σ) | 1-Φ((k-μ)/σ) |
| Normal (with CC) | Φ((k+0.5-μ)/σ)−Φ((k-0.5-μ)/σ) | Φ((k+0.5-μ)/σ) | 1-Φ((k-0.5-μ)/σ) |
Frequently Asked Questions
When is the normal approximation valid for binomial?
When np ≥ 5 and n(1-p) ≥ 5. Both conditions must hold for a good approximation.
What is continuity correction?
Discrete distributions have probability mass at integers. The normal is continuous. Adding ±0.5 aligns boundaries: P(X≤k) uses k+0.5 so the area under the normal from -∞ to k+0.5 approximates the discrete CDF at k.
Why does Poisson have mean = variance?
The Poisson distribution has E(X)=λ and Var(X)=λ. So the normal approximation uses N(λ, λ) — same mean and variance.
When to use hypergeometric vs binomial?
Binomial: sampling with replacement or infinite population. Hypergeometric: sampling without replacement from finite N. If n/N is small, binomial approximates hypergeometric.
Which is better: with or without continuity correction?
With continuity correction is almost always more accurate, especially for P(X=k) and tail probabilities.
Validity Rules by the Numbers
Official Data Sources
Disclaimer: Normal approximation is an approximation. When validity conditions fail (e.g., np < 5), use exact methods. For critical applications, verify with established statistical software.
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