Normal Approximation to the Binomial Calculator
Approximate a binomial probability with a normal curve — with the continuity correction, the exact answer for comparison, and a check on whether the approximation is valid at all.
For large n the binomial is close to a normal with mean np and variance np(1−p).
How the normal approximation to the binomial calculator works
For large n the binomial is close to a normal with mean np and variance np(1−p). The continuity correction — extending the range by half an integer either side — is what makes a discrete count fit a continuous curve.
The usual validity rule is np ≥ 5 and n(1−p) ≥ 5. The calculator computes the exact binomial too, so you can see the error rather than trust the rule.
Formula: z = (k ± 0.5 − np) / √(np(1−p))
Worked examples
| Inputs | P(X ≤ k), normal approximation | Note |
|---|---|---|
| n = 100, p = 30%, k = 35 | 88.496879% | the approximation is accurate to about 0.03 points |
| At the mean | 54.344201% | just above 50% |
| A rare event | 93.809387% | np = 1 — the rule fails and the error shows |
FAQFrequently asked questions
When is the normal approximation valid?
The usual rule is np ≥ 5 and n(1−p) ≥ 5. Some texts demand 10. Either way, the approximation degrades as p moves toward 0 or 1.
What is the continuity correction?
Adding or subtracting 0.5 to the boundary, because a discrete count is being covered by a continuous curve. It typically cuts the error several-fold.
Why bother when the exact answer is computable?
Historically, because the exact sum was laborious. Today it matters mainly for understanding: it is why proportions have normal confidence intervals at all.
What if p is very small?
Use the Poisson approximation instead, with λ = np. It handles rare events far better than the normal.
Is this the central limit theorem?
A special case of it. A binomial count is a sum of independent Bernoulli trials, and the CLT says such sums tend to normal.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- National curriculum in England — Mathematics — the terms and methods taught in UK schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.