How this instrument works
When the number of trials n is reasonably large and the success probability p isn't too close to 0 or 1, a binomial distribution starts to look a lot like a normal distribution with mean μ = np and standard deviation σ = sqrt(np(1−p)) — a consequence of the De Moivre-Laplace theorem, a special case of the central limit theorem. That similarity lets you approximate binomial probabilities using the much simpler normal CDF instead of summing many individual binomial terms by hand.
Because the binomial distribution is discrete, taking only whole-number values, while the normal distribution is continuous, a small adjustment called the continuity correction is applied before converting to a z-score: instead of using x directly, the calculator uses x + 0.5 when approximating P(X ≤ x). This half-unit nudge accounts for the fact that a discrete value like 'exactly 8' actually occupies the continuous interval from 7.5 to 8.5 under the smooth approximating curve, and skipping it introduces a small but avoidable systematic error.
A common rule of thumb for when this approximation is trustworthy is np ≥ 5 and n(1−p) ≥ 5 — some sources use 10 instead of 5 for a stricter standard — comfortably satisfied in the default example here, with n = 100 and p = 0.5. Well outside those conditions, particularly with a small n or a p very close to 0 or 1, the normal approximation can diverge noticeably from the true binomial probability.
- Enter the number of trials into n.
- Enter the probability of success on each trial into p, between 0 and 1.
- Enter the target value into x — the calculator approximates P(X ≤ x).
- Read μ and σ — the normal distribution's mean and standard deviation standing in for the binomial.
- Read Probability — the continuity-corrected approximate P(X ≤ x).
Worked example — approximating P(X ≤ 60) for Binomial(100, 0.5)
You want P(X ≤ 60) for a binomial distribution with n = 100 trials and p = 0.5 success probability — for instance, 60 or fewer heads in 100 fair-coin flips. Enter 100 into n, 0.5 into p, and 60 into x.
μ = 100 × 0.5 = 50, and σ = sqrt(100 × 0.5 × 0.5) = sqrt(25) = 5. With the continuity correction, z = (60 + 0.5 − 50) / 5 = 10.5 / 5 = 2.1. The approximate probability is P(X ≤ 60) ≈ Phi(2.1) ≈ 0.9821 — about a 98.2% chance of seeing 60 or fewer heads in 100 flips of a fair coin, a close match to the true binomial probability at these comfortably large np and n(1−p) values.
Questions
Why is the continuity correction necessary?
The binomial distribution only produces whole-number outcomes, while the normal distribution approximating it is continuous and assigns probability to every real number in between. Without the correction, approximating P(X ≤ x) directly at x understates the true probability slightly, because it ignores the half-unit of area that properly belongs to x under the discrete-to-continuous mapping. Adding 0.5 before computing z corrects for this and noticeably improves accuracy, especially for smaller n.
When is the normal approximation reliable?
A widely used rule of thumb requires both np ≥ 5 and n(1−p) ≥ 5, though some textbooks prefer the stricter threshold of 10, which roughly ensures the binomial distribution isn't too skewed for the symmetric normal curve to approximate it well. When p is very close to 0 or 1, or n is small, the binomial distribution is noticeably lopsided and the normal approximation can be meaningfully off — in those cases, computing the exact binomial probability is more reliable.
Why not just compute the exact binomial probability instead?
You can, and for smaller n it's often more practical to do so directly with modern software. The normal approximation exists mainly for cases where computing the exact binomial CDF is impractical by hand, summing potentially dozens or hundreds of individual binomial terms, or where a fast, easy-to-reason-about estimate is good enough. This calculator's continuity-corrected approximation is typically accurate to a few thousandths once np and n(1−p) are both comfortably above 5.
Does this approximate P(X ≤ x) or P(X < x)?
This calculator specifically approximates P(X ≤ x), applying the continuity correction as x + 0.5. If you instead want P(X < x), a strictly-less-than probability, that's mathematically equivalent to P(X ≤ x−1) for a discrete distribution like the binomial, so enter x−1 as the target value here to get the equivalent result.