Normal Approximation Calculator
Estimate P(a ≤ X ≤ b) for a Binomial(n, p) distribution using the normal approximation. Supports continuity correction and one- or two-sided bounds.
Leave blank for one-sided upper probabilities.
Leave blank for one-sided lower probabilities.
Mean (μ = np): 20.0000
Standard deviation (σ = √np(1 − p)): 3.4641
Adjusted bounds: [14.5, 25.5]
Probability estimate: 88.7649%
z-score (lower): -1.5877
z-score (upper): 1.5877
How to Use This Calculator
- Specify the number of Bernoulli trials and the success probability.
- Choose lower and upper bounds to define the event of interest.
- Decide whether to apply continuity correction to improve accuracy.
- Interpret the approximate probability and z-scores, keeping in mind the normal approximation assumptions.
Formula
Binomial mean: μ = np
Binomial standard deviation: σ = √[np(1 − p)]
Normal approximation: X ≈ N(μ, σ²)
Continuity correction: adjust lower bound by −0.5, upper bound by +0.5
P(a ≤ X ≤ b) ≈ Φ((b + 0.5 − μ) / σ) − Φ((a − 0.5 − μ) / σ)
Full Description
For large n, the binomial distribution is well-approximated by a normal distribution with matching mean and variance. Continuity correction improves accuracy by aligning discrete probabilities with continuous areas under the curve.
The approximation performs best when np and n(1 − p) are both at least 5–10. For extreme probabilities or small n, consider exact binomial calculations.
Frequently Asked Questions
When is the normal approximation acceptable?
Rules of thumb require np ≥ 5 and n(1 − p) ≥ 5. Larger values yield better accuracy.
Why use continuity correction?
Because the binomial distribution is discrete, adjusting boundaries by 0.5 aligns the discrete probability mass with the continuous normal curve.
Can I compute one-tailed probabilities?
Yes. Leave one bound blank to obtain P(X ≤ b) or P(X ≥ a).
How accurate is the approximation?
Accuracy depends on n and p. For critical decisions, compare with the exact binomial CDF.