Continuity Correction Calculator

Continuity correction is an adjustment made when a continuous probability distribution (like the normal distribution) is used to approximate a discrete distribution (like the binomial). Since discrete distributions deal with exact integer values while continuous distributions cover ranges, adding or subtracting 0.5 to the boundary values improves the accuracy of the approximation.

You should apply continuity correction whenever you use the normal distribution to approximate binomial probabilities, especially when the sample size n is moderate. A good rule of thumb is to verify that both np ≥ 5 and n(1−p) ≥ 5 before using the normal approximation. For very large n, the correction has a smaller relative effect but still improves precision.

Replace each discrete probability statement with a continuous one by adjusting the boundaries by 0.5. For example, P(X = n) becomes P(n − 0.5 < X < n + 0.5), P(X ≤ n) becomes P(X < n + 0.5), P(X < n) becomes P(X < n − 0.5), P(X ≥ n) becomes P(X > n − 0.5), and P(X > n) becomes P(X > n + 0.5). You then evaluate these using the standard normal distribution.

The central limit theorem (CLT) underpins the entire approach. It states that the sum of a large number of independent random variables tends toward a normal distribution, regardless of the original distribution. For binomial variables, the CLT justifies approximating the distribution with a normal curve when n is sufficiently large, and continuity correction refines that approximation.

For P(X ≤ 60), the continuity correction transforms the statement to P(X < 60.5) in the continuous normal approximation. So the continuity correction factor shifts the upper bound from 60 to 60.5, giving you a more accurate probability estimate from the normal distribution.

No. The continuity correction always adds or subtracts exactly 0.5 from the boundary value. It is a fixed adjustment inherent to the method of bridging discrete and continuous distributions, so it is always ±0.5 and never zero.

With continuity correction applied, the normal approximation is often very close to the exact binomial probability, particularly when np ≥ 5 and n(1−p) ≥ 5. For small n or extreme probabilities (p near 0 or 1), the approximation is less reliable, and you should prefer the exact binomial formula or other methods like the Poisson approximation.

The exact binomial probability is computed directly from the binomial formula P(X = x) = C(n, x) · p^x · (1−p)^(n−x), giving a precise answer. The normal approximation replaces this calculation with a much simpler area under a normal curve, which is faster to compute for large n. Continuity correction narrows the gap between the two, making the approximation more accurate.