Continuity Correction Calculator — Binomial Normal Approximation
The continuity correction bridges the gap between discrete binomial counts and the continuous normal distribution. Enter the number of trials, success probability and your target count, then choose the inequality — the calculator shows the corrected bounds, z-scores and probability.
Inequality type
Normal approximation to the binomial with continuity correction
How does this calculator work?
To approximate P(X = k) for B(n, p) with a normal: use μ = np, σ = √(np(1−p)), and integrate the normal from k−0.5 to k+0.5. For P(X ≤ k), use the upper bound k+0.5; for P(X ≥ k), use the lower bound k−0.5. The ±0.5 shift is the continuity correction.
Formula
How this is calculated
The binomial distribution B(n, p) counts the number of successes in n independent Bernoulli trials. When n is large (rule of thumb: np ≥ 5 and n(1−p) ≥ 5), the distribution can be approximated by a normal with mean μ = np and standard deviation σ = √(np(1−p)). Because the binomial is discrete but the normal is continuous, matching P(X = k) to a strip of area under the bell curve requires a continuity correction of ±0.5.
The correction rules are: P(X = k) ≈ P(k−0.5 ≤ X ≤ k+0.5), P(X ≤ k) ≈ P(X ≤ k+0.5), P(X < k) ≈ P(X ≤ k−0.5), P(X ≥ k) ≈ P(X ≥ k−0.5), P(X > k) ≈ P(X ≥ k+0.5). Without the correction, probabilities for tails can be noticeably off (a few percent) even at moderate n.
This calculator converts the corrected bounds to z-scores ((bound − μ) / σ), then evaluates the standard normal CDF using the Abramowitz & Stegun polynomial approximation (error < 7.5 × 10⁻⁸). It also shows the uncorrected probability for comparison. For exact binomial probabilities, use a dedicated binomial probability calculator.
Frequently asked questions
Always apply it when approximating a discrete binomial (or any discrete distribution) with the normal. Without it, one-sided probabilities can be off by 1–3 percentage points at n = 30, though the error shrinks as n grows. At very large n (thousands), the correction is negligible.
The common rule of thumb is np ≥ 5 and n(1 − p) ≥ 5. For example, n = 100, p = 0.4 gives np = 40 and n(1−p) = 60 — well within the rule. For extreme probabilities (p close to 0 or 1), larger n or a Poisson approximation may be better.
An exact calculation sums the binomial PMF directly: P(X = k) = C(n,k) × p^k × (1−p)^(n−k). The normal approximation with continuity correction is an approximation — it is fast and gives good intuition but is not exact. Use exact methods when precision matters or when software makes them equally convenient.
TG we-Calculate Editorial Team. (2026). Continuity Correction Calculator — Binomial Normal Approximation [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/continuity-correction-calculator
TG we-Calculate Editorial Team. "Continuity Correction Calculator — Binomial Normal Approximation." TG we-Calculate. 2026. https://we-calculate.com/calculator/continuity-correction-calculator.
TG we-Calculate Editorial Team, "Continuity Correction Calculator — Binomial Normal Approximation," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/continuity-correction-calculator
@misc{wecalculate_continuity_correction_calculator, title = {Continuity Correction Calculator — Binomial Normal Approximation}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/continuity-correction-calculator}}, year = {2026}, note = {TG we-Calculate} }
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