Normal Approximation to the Binomial Calculator

What is the normal approximation to the binomial distribution?

The normal approximation to the binomial distribution replaces a binomial distribution with a normal distribution that has the same mean and standard deviation. If X counts successes in n independent trials, each with success probability p, then X is approximately normal with mean μ = np and standard deviation σ = √(np(1−p)). The approximation is a bridge between two of the most used distributions in statistics: it lets you estimate a binomial probability, such as the chance of at least 60 heads in 100 coin tosses, from the bell curve instead of adding up dozens of binomial terms.

It works because of the central limit theorem: as the number of trials grows, the shape of the binomial distribution becomes more and more like the normal curve. The calculator above makes this visible — the bars are the exact binomial probabilities and the smooth curve is the normal approximation laid over them. When n is large and p is not too close to 0 or 1, the two are almost the same. Use our normal distribution calculator to work directly with the bell curve, or the standard normal distribution page to see how z-scores turn any normal problem into a standard one.

When can you use the normal approximation to the binomial?

Use the normal approximation only when both np ≥ 5 and n(1−p) ≥ 5. These two checks make sure the binomial distribution is not too skewed for the symmetric bell curve to describe it. With n = 100 and p = 0.5, np = 50 and n(1−p) = 50, so the approximation is excellent. With n = 50 and p = 0.2, np = 10 and n(1−p) = 40, so it is still fine. But with n = 10 and p = 0.1, np = 1, which is far below 5 — the distribution is bunched up near zero and the bell curve is a poor fit, so the approximation is not recommended and the exact binomial probability should be used instead.

As a rule of thumb, the approximation gets better as n grows and as p gets closer to 0.5. It is at its worst when p is near 0 or 1, because the binomial distribution is then strongly skewed. The empirical rule probability finder shows the same idea from the other side: it finds the probability inside a range of a normal curve, which is exactly what the approximation does once the binomial has been replaced by its normal twin. If you are testing a claim about a single proportion, the one-proportion z-test calculator uses the same np and n(1−p) checks before it trusts the normal curve.

How do you calculate a normal approximation to a binomial probability?

Follow four steps. First find the mean μ = np and the standard deviation σ = √(np(1−p)). Then apply the continuity correction: move the boundary half a unit towards the middle of the range you want, because the binomial counts whole numbers while the normal curve is continuous. Next turn the corrected boundary into a z-score with z = (k ± 0.5 − np) / √(np(1−p)). Finally read the probability from the standard normal curve, Φ(z) — or from the calculator above, which does all four steps and shows the exact binomial value beside the approximation.

Worked example: 100 fair coin tosses, p = 0.5, and you want P(X ≥ 60), the chance of at least 60 heads. The mean is μ = 100 × 0.5 = 50 and the standard deviation is σ = √(100 × 0.5 × 0.5) = 5. The continuity correction moves the boundary from 60 to 59.5, giving z = (59.5 − 50) / 5 = 1.90. Then P(X ≥ 60) ≈ 1 − Φ(1.90) ≈ 1 − 0.9713 = 0.0287. The exact binomial probability is 0.0284, so the absolute error is only 0.0003. If you need the z-score on its own, try the z-score calculator, and the guide to how to calculate a z-score explains the same formula step by step.

Worked examples

Example 1: at least 60 heads in 100 tosses

n = 100, p = 0.5, P(X ≥ 60). Mean μ = 50, standard deviation σ = 5. With the continuity correction, z = (59.5 − 50) / 5 = 1.90, so the approximate probability is 1 − Φ(1.90) ≈ 0.0287. The exact binomial probability is 0.0284, giving an absolute error of 0.0003. The approximation is good because np = 50 and n(1−p) = 50 are both well above 5.

Example 2: exactly 6 successes in 20 trials

n = 20, p = 0.3, P(X = 6). Mean μ = 6, standard deviation σ = √(20 × 0.3 × 0.7) ≈ 2.05. For an exact value, the continuity correction uses the interval from 5.5 to 6.5, giving z-scores of −0.24 and 0.24. The approximate probability is Φ(0.24) − Φ(−0.24) ≈ 0.1927, against an exact binomial probability of 0.1916 — an absolute error of 0.0011. Without the correction the approximation would be much worse, because a single value is the hardest case for a continuous curve.

Example 3: at most 8 successes in 50 trials

n = 50, p = 0.2, P(X ≤ 8). Mean μ = 10, standard deviation σ = √(50 × 0.2 × 0.8) ≈ 2.83. The continuity correction moves the boundary to 8.5, giving z = (8.5 − 10) / 2.83 ≈ −0.53, so the approximate probability is Φ(−0.53) ≈ 0.2979. The exact binomial probability is 0.3073, so the absolute error is 0.0094 — larger than in the first two examples, because n = 50 is smaller and p = 0.2 puts the event in the lower tail. The approximation is still valid here: np = 10 and n(1−p) = 40 are both at least 5.

Normal approximation vs exact binomial probability

The table below compares the two for the worked examples. The approximation is close whenever the validity checks pass, and the continuity correction is what keeps the error small — especially for "exactly" events, where the uncorrected approximation can be badly off.

Notice the pattern: the error is smallest when the event is in the middle of the distribution and largest in the tails or for single values. If you need a p-value from a z-score after the approximation, the p-value calculator converts it directly. For more worked bell-curve problems, see the normal distribution examples.

What does the graph show?

The graph at the top of the page draws the two distributions together. Each bar is the exact binomial probability P(X = k) for one value of k. The smooth curve is the normal distribution with the same mean and standard deviation, and the shaded part of the curve is the event you asked for — the area that gives the approximate probability. When the bars follow the curve closely, the approximation is trustworthy; when they pull away from it, especially near 0 or n, that is the visual sign that the validity checks are failing.