What this calculator does
A continuity correction is the small ±0.5 adjustment applied to a discrete value before it is converted into a z-score, whenever a discrete distribution (like the binomial or Poisson) is being approximated by a continuous one (the normal curve). Without it, a discrete count such as "12 successes" gets treated as a single infinitely thin point on a continuous scale, which biases the resulting probability estimate.
The correction works by widening that single point into a small interval: to estimate P(X ≤ x), use x + 0.5 as the cutoff on the normal curve; for P(X ≥ x), use x − 0.5; and for the probability of an exact value P(X = x), bracket it between x − 0.5 and x + 0.5. This calculator shows the corrected z-score alongside the uncorrected one, so the size of the adjustment is visible rather than hidden inside a single answer.
The formula
Enter the discrete value, the mean and the standard deviation of the distribution being approximated. Choose which direction the correction should run: adding 0.5 for a "less than or equal to" probability, subtracting 0.5 for a "greater than or equal to" probability, or bracketing both sides for the probability of an exact value. The calculator returns the corrected z-score, the resulting probability estimate under the normal curve, and the same figures without the correction for comparison.
| Term | Meaning |
|---|---|
| Continuity correction | The ±0.5 adjustment made to a discrete value before treating it as a point on a continuous normal curve. |
| z-score | How many standard deviations the (corrected) value sits from the mean: (x − μ) / σ. |
| μ and σ | The mean and standard deviation of the distribution being approximated, such as those from a binomial or Poisson distribution. |
The inputs explained
| Field | What to enter |
|---|---|
| Discrete value (x) | The discrete value (a count, such as a number of successes) being converted to a z-score. |
| Mean (μ) | The mean of the distribution being approximated by the normal curve. |
| Standard deviation (σ) | The standard deviation of that distribution. Must be greater than zero. |
| Correction direction | Which probability direction the correction applies to: at-or-below, at-or-above, or an exact single value. |
When to use it
Approximating a binomial probability
When np and n(1 − p) are both comfortably above 5, a binomial probability such as P(X ≤ 12 successes) is commonly estimated with the normal curve. Applying the continuity correction to that cutoff before computing the z-score noticeably improves the estimate compared with skipping it.
Checking how much the correction actually changes the answer
The gap between the corrected and uncorrected probability shrinks as the standard deviation grows relative to the ±0.5 adjustment. Comparing both figures side by side shows when the correction is worth bothering with and when it barely moves the result.
Estimating the probability of one specific count
For the probability of hitting an exact discrete value, such as P(X = 10), the correction brackets that value between x − 0.5 and x + 0.5 rather than trying to evaluate a single point on a continuous curve, which would otherwise always come out to zero.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How the correction changes the estimate as x moves away from the mean
A fixed mean of 10 and standard deviation of 2.5, with the discrete value x moving further from the mean each row.
| Discrete value x | Continuity-corrected z-score | Uncorrected z-score (no adjustment) | Difference the correction makes |
|---|---|---|---|
| 8 | -0.600 | -0.800 | 0.200 in z, 6.240 percentage points in probability |
| 10 | 0.200 | 0.000 | 0.200 in z, 7.926 percentage points in probability |
| 12 | 1.000 | 0.800 | 0.200 in z, 5.320 percentage points in probability |
| 14 | 1.800 | 1.600 | 0.200 in z, 1.887 percentage points in probability |
| 16 | 2.600 | 2.400 | 0.200 in z, 0.354 percentage points in probability |
Questions
When is a continuity correction actually needed?
Only when a discrete distribution, such as the binomial or Poisson, is being approximated by the continuous normal distribution. If the underlying data is already continuous, there is nothing to correct for.
Why is it ±0.5 specifically?
A discrete value like 12 really represents the whole interval from 11.5 to 12.5 once it is mapped onto a continuous scale. Shifting the cutoff by half a unit accounts for that interval rather than treating 12 as an infinitely thin point.
Does the correction matter for large standard deviations?
Less so. The fixed 0.5 adjustment becomes a smaller fraction of a standard deviation as σ grows, so its effect on the resulting z-score and probability shrinks correspondingly.
How does this relate to the normal approximation of a binomial distribution?
The normal approximation calculator derives μ and σ directly from n and p and can optionally apply this same correction internally. This calculator is for working with an arbitrary mean and standard deviation directly, or for seeing the corrected and uncorrected results compared side by side.
To derive μ and σ for a binomial distribution before applying this correction, see the normal approximation calculator. For a general z-score without any discrete-to-continuous adjustment, use the z-score calculator.