continuitycorrection

Continuity correction, also known as a correction for continuity, is a technique used when approximating a discrete distribution by a continuous distribution, typically the normal distribution. The idea is to adjust boundary values by a half unit to account for the fact that the discrete variable takes distinct integer values, while the continuous approximation is smooth.

In practice, when X is discrete and approximated by a normal distribution, probabilities are computed with boundaries shifted by 0.5. For example, if X ~ Bin(n, p) is approximated by a normal with mean μ = np and variance σ² = npq, then:

- P(X ≤ k) ≈ Φ((k + 0.5 − μ) / σ)

- P(X ≥ k) ≈ 1 − Φ((k − 0.5 − μ) / σ)

- P(X = k) ≈ Φ((k + 0.5 − μ) / σ) − Φ((k − 0.5 − μ) / σ)

Similarly, continuity corrections are used when Poisson or other discrete distributions are approximated by a normal distribution.

Uses and variants include improving normal approximations in hypothesis tests and confidence intervals for proportions or counts, and in some 2×2 contingency table analyses (where Yates’ correction is a related, though distinct, form). The correction tends to improve accuracy for moderate sample sizes or probabilities near 0 or 1, particularly when the discrete step size is nontrivial.

Limitations include that the improvement is not universal and becomes negligible for large samples or when exact discrete calculations are feasible. In modern practice, the correction is one of several tools available for balancing accuracy and computation.