Normal Approximation to Binomial Calculator
Compare an exact binomial probability with its normal approximation using the continuity correction, z-scores and the np ≥ 5 rule, and see the exact error.
e.g. 0.3, 30% or 3/10
Step-by-step solution
About this calculator
For large n, a binomial distribution looks very much like a normal (bell-shaped) curve with the same mean and standard deviation. Before computers, this let people estimate binomial probabilities with a z-table, and it is still the basis of the common z-test for a proportion.
This calculator shows both answers side by side: the exact binomial probability, and the normal approximation with and without the continuity correction. It also shows the z-score, checks the usual rule of thumb (np ≥ 5 and n(1 − p) ≥ 5), and reports how far the approximation is from the exact value, so you can see when the shortcut is safe.
How to use it
- Enter n, p and k.
- Choose the event: P(X ≤ k), P(X < k), P(X ≥ k), P(X > k) or P(X = k).
- Read the exact probability, the approximation, the corrected boundary and z-score, and the error.
- Check the rule-of-thumb badge. If it fails, trust the exact value.
The formulas
Matching mean and standard deviation
μ = n·p σ = √(n·p·(1 − p))The approximating normal curve uses exactly the binomial mean and standard deviation.
Continuity correction
P(X ≤ k) ≈ P(Y < k + 0.5) P(X ≥ k) ≈ P(Y > k − 0.5) P(X = k) ≈ P(k − 0.5 < Y < k + 0.5)The binomial only takes whole-number values, and each one is like a bar of width 1 centred on k. Moving the boundary half a unit makes the smooth curve cover the whole bar instead of cutting it in half.
z-score
z = (x − μ) / σ then P(Y < x) = Φ(z)Standardise the corrected boundary x and look up the standard normal cumulative probability Φ(z). For “greater than” events use 1 − Φ(z).
Rule of thumb
n·p ≥ 5 and n·(1 − p) ≥ 5Both expected counts should be large enough that the distribution is not squashed against 0 or n. Some textbooks use 10 instead of 5 for a stricter check.
Worked examples
Example 1: 55 or fewer heads in 100 flips
A fair coin is flipped 100 times. Estimate P(X ≤ 55) with the normal approximation and compare it with the exact answer.
- μ = 100 × 0.5 = 50, σ = √(100 × 0.5 × 0.5) = 5. Rule check: np = 50 and n(1 − p) = 50, both ≥ 5.
- Continuity correction: P(X ≤ 55) ≈ P(Y < 55.5).
- z = (55.5 − 50) / 5 = 1.1, so the approximation is Φ(1.1) = 0.864334.
- Exact binomial: P(X ≤ 55) = 0.864373.
- Without the correction, z = (55 − 50) / 5 = 1 and Φ(1) = 0.841345 — noticeably worse.
Answer: The corrected approximation is off by only 4 × 10⁻⁵; skipping the correction makes the error about 0.023.
Example 2: when the rule of thumb fails
A process makes 10% defective items. In a sample of 20, what is P(X ≤ 1)?
- μ = 20 × 0.1 = 2, σ = √(20 × 0.1 × 0.9) ≈ 1.342. Rule check: np = 2, which is below 5.
- Corrected boundary 1.5, z = (1.5 − 2) / 1.342 = -0.3727, approximation Φ(z) = 0.354694.
- Exact binomial: P(X ≤ 1) = 0.9²⁰ + 20 × 0.1 × 0.9¹⁹ = 0.391747.
Answer: The approximation is off by 0.037 (about 9% of the true value). The distribution is skewed when np is small, so a symmetric bell curve fits poorly — use the exact value here.
When to use the normal approximation to binomial calculator
- The situation is binomial: fixed n, two outcomes, constant p and independent trials.
- n·p and n·(1 − p) are both at least 5 (or 10 for a stricter standard).
- You need a quick estimate by hand, are learning how z-tests for proportions work, or want to check a textbook answer.
- If software is available, the exact binomial probability is always at least as accurate; the approximation is a convenience, not an improvement. For large n with many successes, the binomial distribution calculator handles the exact sum directly.
When it does not apply
- Small n or extreme p (close to 0 or 1). When n is large but p is small, the Poisson distribution with mean n·p is usually a better approximation.
- Probabilities far out in the tails, such as P(X ≥ 90) for 100 coin flips. The absolute error may be tiny, but the relative error can be huge.
Common mistakes
- Forgetting the continuity correction, or moving the boundary the wrong way. “At most k” uses k + 0.5; “at least k” uses k − 0.5; “fewer than k” is “at most k − 1”, so it uses k − 0.5.
- Using the variance n·p·(1 − p) where the standard deviation √(n·p·(1 − p)) belongs in the z-score.
- Applying the approximation when n·p or n·(1 − p) is small, where it can be badly off (see Example 2). The binomial distribution calculator gives the exact value for any n up to 100,000.
- Approximating P(X = k) without a correction: a continuous curve gives exactly 0 for a single point, so you must use the interval from k − 0.5 to k + 0.5.
Frequently asked questions
What is the continuity correction?
An adjustment of 0.5 to the boundary when a continuous normal curve approximates a discrete distribution. Each whole number k is treated as the interval from k − 0.5 to k + 0.5, so P(X ≤ k) becomes P(Y < k + 0.5).
When can I use the normal approximation to the binomial?
When n·p ≥ 5 and n·(1 − p) ≥ 5 (some courses require 10). Under those conditions the binomial is close enough to symmetric that the bell curve fits well.
How accurate is the normal approximation?
It depends on n, p and the event. For 100 fair coin flips, P(X ≤ 55) is off by about 4 × 10⁻⁵ with the correction. For n = 20 and p = 0.1 the error is about 0.037. The calculator shows the exact error for your inputs.
Should I use + 0.5 or − 0.5?
Widen the region to include the whole bar of every value you want. For X ≤ k or X > k the boundary is k + 0.5; for X ≥ k or X < k it is k − 0.5.
Why use the approximation if the exact answer is available?
Mainly for hand calculation, exams, and because many statistical tests for proportions are built on it. When you can compute the exact binomial probability, as this page does, prefer the exact value.