How this calculator works
A binomial count has mean np and variance np(1 - p). Approximate its discrete mass using a normal curve with those moments. Continuity correction replaces an inclusive integer boundary by the corresponding half-integer boundary. Exactly k uses the interval k - 0.5 to k + 0.5; without correction a single point has zero normal probability. The approximation may be poor for rare events, small samples or distant tails. Trial outcomes must be independent with a common success probability.
Formula
μ = np; σ = √(np(1 - p)); z = (boundary - μ)/σ. P(X = k) ≈ Φ((k + 0.5 - μ)/σ) - Φ((k - 0.5 - μ)/σ).
Worked example
For n = 100 and p = 0.5, the mean is 50 and SD is 5. The corrected probability of exactly 50 is Φ(0.1) - Φ(-0.1) ≈ 0.07965567. The uncorrected probability of at most 50 is 0.5.
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Frequently asked questions
Why add or subtract 0.5?
An integer count represents a unit-wide bin. The half-unit boundary makes the continuous area better match that bin.
Can this replace exact binomial tails everywhere?
No. A good central approximation can still have large relative error in a far tail.
What if the success probability is zero or one?
The count is deterministic and its normal standard deviation is zero. This page requires a non-degenerate probability.