Normal Continuity Correction Calculator
The normal distribution is a convenient approximation to the binomial when the number of trials is large. Because the binomial is discrete and the normal is continuous, a continuity correction of half a unit makes the approximation noticeably more accurate. This calculator approximates P(X is at most k) for a binomial with n trials and success probability p. It reports the mean, standard deviation, the corrected z-score using cutoff k + 0.5, and the resulting probability, alongside the uncorrected value so you can see the difference the correction makes.
Continuity correction formula
mean mu = n * p
std deviation sigma = sqrt(n * p * (1 - p))
corrected z = (k + 0.5 - mu) / sigma
P(X <= k) approx = Phi(z), the standard normal CDF
Phi is the standard normal cumulative distribution function, evaluated here with a high-accuracy error-function approximation. The uncorrected version uses cutoff k instead of k + 0.5.
Using the result
- The mean and standard deviation come directly from the binomial parameters n and p.
- The correction adds 0.5 to k so the normal area includes the bar at k.
- The approximation is most accurate when n*p and n*(1-p) are both at least 5.
- Probabilities are clamped between 0 and 1 by the normal CDF.
- p must lie strictly between 0 and 1 and n must be positive.
Continuity correction: frequently asked questions
What is a continuity correction?
When a discrete binomial distribution is approximated by a continuous normal distribution, each integer is treated as covering the interval half a unit on either side. The continuity correction adds or subtracts 0.5 to the cutoff so the normal area matches the discrete probability more closely.
How is P(X less than or equal to k) corrected?
For the binomial probability P(X <= k), the continuity correction uses the cutoff k + 0.5 before applying the normal approximation: P(X <= k) is approximated by the normal probability below k + 0.5, standardised as z = (k + 0.5 - n*p) / sqrt(n*p*(1-p)).
When is the normal approximation appropriate?
A common rule of thumb is that both n*p and n*(1-p) should be at least 5 (some texts use 10). When this holds, the binomial is roughly symmetric and the normal approximation with continuity correction is reasonably accurate.
What is the mean and standard deviation used?
For a binomial with n trials and success probability p, the mean is n*p and the standard deviation is sqrt(n*p*(1-p)). These are the parameters of the matching normal distribution before the continuity correction is applied.
Why does the corrected answer differ from the uncorrected one?
Without the correction, the normal area stops exactly at the integer k and misses the half-unit bar that the discrete value k occupies. Adding 0.5 includes that bar, giving a closer match to the true binomial probability, especially for small n.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Normal distribution.
- NIST Digital Library of Mathematical Functions: Error functions.
Reviewed by the CalculatorHub team, edited by James Graham, 16 June 2026. See our methodology.