Continuity Correction Calculator
Apply the continuity correction factor when you need to use the normal distribution to approximate a binomial probability. Enter the number of trials, the probability of success, and the number of successes you are interested in, then choose an inequality type. The calculator shows the corrected interval, the z-score, and the approximated probability, along with a step-by-step walkthrough of the math.
What is continuity correction?
A continuity correction is a small adjustment - adding or subtracting 0.5 - applied when you use a continuous probability distribution (the normal) to approximate a discrete one (the binomial). Because the binomial distribution only takes integer values but the normal distribution is continuous, the correction bridges the gap between the two. Without it, you would undercount or overcount probability mass at the boundary of each integer, and the approximation would be noticeably off, especially for small n.
When can you use the normal approximation?
The rule of thumb is that both np and n(1-p) must be at least 5 before the approximating normal distribution is a reliable stand-in for the exact binomial. Here n is the number of trials and p is the probability of success. When this condition holds, the binomial distribution is close to bell-shaped and the correction makes the approximation accurate to within a fraction of a percent in most cases. If the condition fails - particularly when p is very close to 0 or 1, or n is very small - you should use an exact binomial calculator instead.
How to apply the correction: the five rules
The adjustment depends on the inequality sign. For P(X = x), replace the single point with the interval (x - 0.5, x + 0.5). For P(X at most x), shift the bound up to x + 0.5. For P(X strictly less than x), shift it down to x - 0.5. For P(X at least x), shift the bound down to x - 0.5. For P(X strictly greater than x), shift the bound up to x + 0.5. The resulting continuous bound is then standardised into a z-score using z = (bound - mu) / sigma, and the standard normal table gives the probability.
Step-by-step worked example
Suppose you flip a fair coin 100 times and want P(X at most 43). Step 1: check the condition - np = 50 and n(1-p) = 50, both well above 5. Step 2: compute the mean mu = 100 * 0.5 = 50 and the standard deviation sigma = sqrt(100 * 0.5 * 0.5) = 5. Step 3: apply the correction - "at most 43" becomes "X < 43.5". Step 4: standardise - z = (43.5 - 50) / 5 = -1.30. Step 5: look up the standard normal CDF - Phi(-1.30) is approximately 0.0968. The exact binomial result is 0.0967, so the corrected approximation is accurate to three decimal places.
Continuity correction rules
| Binomial statement | Corrected continuous form | Direction |
|---|---|---|
| P(X = x) | P(x - 0.5 < X < x + 0.5) | Two-tailed slice |
| P(X <= x) | P(X < x + 0.5) | Left tail |
| P(X < x) | P(X < x - 0.5) | Left tail |
| P(X >= x) | P(X > x - 0.5) | Right tail |
| P(X > x) | P(X > x + 0.5) | Right tail |
How to adjust each binomial probability statement before using the normal approximation.
Frequently asked questions
Why do we add or subtract 0.5 in the continuity correction?
The binomial distribution assigns all of the probability for the outcome "X = k" to the single integer k. When you represent this with a continuous normal distribution, the equivalent region is the strip from k - 0.5 to k + 0.5, since that strip captures the same probability mass under the smooth curve. The 0.5 adjustment is therefore not arbitrary - it comes directly from splitting the gap between consecutive integers.
Does the continuity correction always improve accuracy?
In the vast majority of cases, yes. The corrected approximation is almost always closer to the exact binomial probability than the uncorrected version. The improvement is most pronounced when n is moderate (roughly 10 to 50) and p is not too extreme. For very large n the improvement shrinks because the normal curve is already a near-perfect fit without any correction.
What is the condition for the normal approximation to be valid?
Both np and n(1-p) must be at least 5. Some textbooks use a stricter threshold of 10. The condition ensures the binomial distribution is sufficiently symmetric and bell-shaped. When p is very close to 0 or 1, even large n values may not satisfy the condition, because the distribution is skewed. In those cases, use an exact binomial probability calculator.
Is the continuity correction used in practice today?
Mostly for teaching. Modern software computes exact binomial probabilities instantly, making the normal approximation unnecessary for routine calculations. However, the correction still appears in introductory statistics courses because it illustrates the relationship between discrete and continuous distributions and explains why the normal distribution arises so naturally in probability theory.
What is the difference between P(X < x) and P(X <= x) after correction?
For a discrete distribution, these differ by exactly one probability mass, P(X = x). After the continuity correction, "X at most x" maps to the continuous interval "X < x + 0.5", while "X strictly less than x" maps to "X < x - 0.5". The difference between the two corrected bounds is exactly 1 unit, which corresponds to removing the x + 0.5 to x + 0.5 slice - consistent with removing P(X = x) from the sum.