Normal Approximation to Binomial Calculator
When n is large, the Binomial(n, p) distribution is well approximated by a Normal(np, np(1−p)) distribution. Enter n, p and a target count k, choose the probability type, and get the approximated probability with a bell-curve diagram.
Probability type
Normal approximation with continuity correction applied
- 1
Mean μ = np
100 × 0.4 = 40 - 2
Standard deviation σ = √(np(1−p))
√(40 × (1 − 0.4)) = 4.899 - 3
z with continuity correction
(35 + 0.5 − 40) ÷ 4.899 = -0.9186 - 4
P(X ≤ k) = Φ(z)
Φ(-0.9186) = 0.179163Φ is the standard normal CDF; continuity correction adds 0.5 to k.
How does this calculator work?
Binomial(n, p) ≈ Normal(μ=np, σ=√(np(1−p))) when np ≥ 5 and n(1−p) ≥ 5. Apply continuity correction: P(X ≤ k) ≈ Φ((k+0.5−μ)/σ). The calculator shows the bell curve with the shaded probability region for at-most, at-least, or exactly k successes.
Formula
How this is calculated
The binomial distribution B(n, p) describes the number of successes in n independent trials each with probability p. Computing exact binomial probabilities for large n requires summing many terms, which is slow and prone to overflow. When np ≥ 5 and n(1−p) ≥ 5, the Central Limit Theorem guarantees the binomial is well approximated by a normal distribution with mean μ = np and standard deviation σ = √(np(1−p)).
The continuity correction improves accuracy by treating the discrete value k as the continuous interval [k−0.5, k+0.5]. For P(X ≤ k) we shade up to k+0.5, giving z = (k+0.5−μ)/σ; for P(X ≥ k) we shade from k−0.5, giving z = (k−0.5−μ)/σ; for P(X = k) we take the area between k−0.5 and k+0.5. Without the correction the results can be noticeably off, especially near the tails or when n is moderate.
For proportions, dividing by n gives: sample proportion p̂ ≈ Normal(p, p(1−p)/n), which underpins confidence intervals and hypothesis tests for proportions. The approximation is reliable when np ≥ 5 and n(1−p) ≥ 5; if either condition fails, use the exact binomial (or a Poisson approximation when p is very small).
Frequently asked questions
The common rule of thumb is np ≥ 5 AND n(1−p) ≥ 5. Some texts use 10 for more accuracy. The approximation improves with larger n and p near 0.5. When p is very small (rare events), the Poisson approximation Poisson(λ = np) is usually better.
The binomial is discrete (integer values) while the normal is continuous. Treating the integer k as the interval [k−0.5, k+0.5] before computing the normal CDF is the continuity correction. It reduces approximation error substantially, especially for moderate n — without it, P(X = k) would always be 0 under the continuous distribution.
For a proportion p̂ = X/n, the approximation gives a z-score: z = (p̂ − p₀) / √(p₀(1−p₀)/n). This is the one-proportion z-test statistic. The P-value is then read from the normal CDF — which is exactly what this calculator computes for the count X = k = n × p̂.
Also known as
TG we-Calculate Editorial Team. (2026). Normal Approximation to Binomial Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/normal-approximation-calculator
TG we-Calculate Editorial Team. "Normal Approximation to Binomial Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/normal-approximation-calculator.
TG we-Calculate Editorial Team, "Normal Approximation to Binomial Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/normal-approximation-calculator
@misc{wecalculate_normal_approximation_calculator, title = {Normal Approximation to Binomial Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/normal-approximation-calculator}}, year = {2026}, note = {TG we-Calculate} }
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