Normal Approximation to the Binomial

Understanding the normal approximation to the binomial — concept and principle

A binomial variable X counts successes in n independent trials, each with the same success probability p: defective parts in a batch, voters who answer yes, free throws made. Exact binomial probabilities require summing many terms, so for large n they are often approximated with a normal curve. This calculator gives P(X ≤ x) and P(X ≥ x) with a continuity correction and checks whether the approximation is reasonable.

How it is calculated

1. Mean μ = np and standard deviation σ = √(np(1 − p)).
2. P(X ≤ x) ≈ Φ((x + 0.5 − μ) ÷ σ).
3. P(X ≥ x) ≈ 1 − Φ((x − 0.5 − μ) ÷ σ).
4. Validity check: the approximation is labelled acceptable when both np ≥ 10 and n(1 − p) ≥ 10.

Φ is the standard normal cumulative distribution function. n is a whole number of trials, p a probability between 0 and 1, and x a whole number of successes.

The principle

A binomial count is a sum of n independent 0/1 outcomes, so by the central limit theorem (in this case the de Moivre–Laplace theorem) its distribution approaches a normal curve with the same mean and variance as n grows. The binomial is discrete, while the normal is continuous: the probability of exactly x is represented by the area of a bar from x − 0.5 to x + 0.5. "At most x" therefore runs up to x + 0.5, and "at least x" starts at x − 0.5. This ±0.5 is the continuity correction, and it noticeably improves accuracy.

Worked example

1. Inputs: n = 100, p = 0.3, x = 35.
2. μ = 100 × 0.3 = 30; σ = √(100 × 0.3 × 0.7) = √21 ≈ 4.5826.
3. P(X ≤ 35): z = (35.5 − 30) ÷ 4.5826 ≈ 1.2002, Φ(1.2002) ≈ 0.8850.
4. P(X ≥ 35): z = (34.5 − 30) ÷ 4.5826 ≈ 0.9820, 1 − Φ(0.9820) ≈ 0.1631.
5. np = 30 and n(1 − p) = 70 are both at least 10, so the check says yes.

How to read the result

Here there is about an 88.5% chance of 35 or fewer successes and about a 16.3% chance of 35 or more. The two add up to more than 1 because both include exactly 35. The exact binomial values are 0.8839 and 0.1629, so the approximation is accurate to about 0.001. If the check reports that the approximation is rough, use exact binomial probabilities or, for small p and large n, a Poisson approximation.

Common mistakes

• Omitting the continuity correction, or applying it in the wrong direction.
• Using the approximation when p is close to 0 or 1 with a small n, where the binomial is strongly skewed.
• Using it when trials are not independent or p changes from trial to trial.

Frequently asked questions

Why do some textbooks use np ≥ 5 instead of 10?

Both are rules of thumb; 10 is the stricter, more common modern guideline.

How do I get P(X = x)?

Approximate it as Φ((x + 0.5 − μ) ÷ σ) − Φ((x − 0.5 − μ) ÷ σ), which is P(X ≤ x) + P(X ≥ x) − 1.

Is this still needed when software can compute exact values?

Exact values are preferable, but the approximation remains useful for quick estimates and underlies the usual tests for proportions.

Sources: Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 1 (Wiley). · NIST/SEMATECH e-Handbook of Statistical Methods.