Continuity Correction Calculator

Approximate binomial probabilities with a normal distribution and the correct half-unit boundary adjustment.

Normal Approximation to the Binomial
Enter a binomial model and event boundary to calculate its continuity-corrected probability.

About continuity correction

A binomial random variable counts successes and therefore takes only whole-number values. A normal random variable is continuous and can take every value on the number line. When a normal curve is used to approximate a binomial distribution, continuity correction shifts each discrete boundary by one half. That small adjustment makes the area under the continuous curve line up more closely with the probability bars of the discrete distribution. For an event such as at most 20 successes, the corrected normal boundary is 20.5 because the binomial bar centered at 20 extends from 19.5 to 20.5. For at least 20, the lower boundary becomes 19.5. Exactly 20 is represented by the interval from 19.5 through 20.5, while an inclusive event from 20 through 30 uses 19.5 and 30.5. The calculator applies these rules automatically before standardizing each finite boundary into a z-score. The approximating normal distribution has mean n times p and standard deviation equal to the square root of n times p times one minus p. After correction, each boundary is converted by subtracting the mean and dividing by the standard deviation. The required probability is then the area between the resulting z-scores. One-sided events use the entire remaining tail, so they need only one finite corrected boundary. A normal approximation is generally most useful when both expected successes, n times p, and expected failures, n times one minus p, are at least 5. Some courses and analysts prefer a threshold of 10 for a more conservative check. If either count is small, the binomial distribution may be strongly skewed and an exact binomial calculation is safer. The approximation also becomes undefined when p is exactly zero or one because the standard deviation is zero. Continuity correction matters most for moderate sample sizes and probabilities near a decision boundary. Its impact usually shrinks as the number of trials grows, but applying it remains a sound way to improve a hand-calculated normal approximation. Use the displayed mean, standard deviation, and probability to check homework, plan an analysis, or compare an approximate answer with statistical software.

Continuity correction examples

These examples show how discrete event boundaries move before the normal probability is evaluated.

Binomial EventCorrected BoundaryInterpretation
n = 100, p = 0.5, X at most 55X below 55.5; probability 0.864334The upper edge of the bar for 55 is included.
n = 50, p = 0.4, X at least 20X above 19.5; probability 0.557469The lower edge shifts down to include all of 20.
n = 80, p = 0.25, X exactly 2019.5 below X below 20.5One discrete bar becomes a one-unit continuous interval.

How to use the continuity correction calculator

  1. Enter the number of independent binomial trials and the probability of success on each trial.
  2. Choose whether the event is at most, at least, exactly, or between two success counts.
  3. Enter the target count and, for a between event, the inclusive upper target.
  4. Select Calculate Probability and read the corrected normal probability, mean, and standard deviation.

Continuity correction FAQ

Why is continuity correction needed?

A binomial count is discrete while a normal model is continuous. Moving each inclusive boundary by half a unit aligns normal areas with the full probability bars.

When should I use a normal approximation?

Use it when expected successes and failures are both reasonably large, commonly at least 5 or 10. For sparse or highly skewed cases, calculate the exact binomial probability instead.

Do I add or subtract 0.5?

An inclusive upper boundary moves up by 0.5, while an inclusive lower boundary moves down by 0.5. An exact count uses both adjustments to create a one-unit interval.

Can p equal zero or one?

Those values describe deterministic outcomes and produce a standard deviation of zero. The normal approximation is therefore undefined, although the exact binomial answer is immediate.

Is the corrected answer exact?

No, it remains an approximation based on the normal distribution. Continuity correction usually improves that approximation but does not replace exact binomial calculations.