What the continuity correction is
The continuity correction is a ±0.5 adjustment you apply when you approximate a discrete distribution with a continuous one — most commonly when the normal distribution is used to approximate the binomial. A binomial count X can only land on whole numbers (0, 1, 2, …), but the normal curve is smooth and defined on every real value. The correction bridges that gap by treating each integer k as the interval from k − 0.5 to k + 0.5 — the width of the histogram bar that sits over k — so the area under the normal curve matches the probability mass of the discrete bar.
The formula
For a binomial random variable X with n trials and success probability p, the mean and standard deviation are:
- Mean: μ = np
- Standard deviation: σ = √(np(1 − p))
The corrected z-score replaces the raw count with a boundary shifted by 0.5 in the direction that keeps the endpoint inside the region you want:
- P(X ≤ k) ≈ Φ((k + 0.5 − μ) / σ)
- P(X < k) ≈ Φ((k − 0.5 − μ) / σ)
- P(X ≥ k) ≈ 1 − 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)
Here Φ is the standard normal cumulative distribution function (the area to the left of a z-score). The rule for the sign: use +0.5 when you want to include a value up to and including k on the low side, and −0.5 when the boundary is on the high side of k. If in doubt, sketch the histogram bars you intend to include and shift the boundary to the outer edge of those bars.
A worked example
Flip a fair coin 100 times (n = 100, p = 0.5). Then μ = 50 and σ = √(100 × 0.5 × 0.5) = √25 = 5. To approximate P(X ≥ 60), shift to 59.5: z = (59.5 − 50)/5 = 1.90, and P(X ≥ 60) ≈ 1 − Φ(1.90) ≈ 0.0287. The exact binomial value is about 0.0284, so the correction lands within a fraction of a percent. Without the correction (using 60 directly), z = 2.00 gives 0.0228 — noticeably too low.
Why the correction matters
Skipping the ±0.5 shift systematically biases the approximation, and the error is largest exactly where people care most: in the tails and for single-value probabilities. Without a correction, P(X = k) collapses to zero (a continuous distribution assigns zero probability to any single point), which is obviously wrong for a discrete count. The correction restores a sensible, positive answer equal to the area of that one histogram bar.
When the normal approximation is valid
The approximation — corrected or not — only works when the binomial is roughly bell-shaped. The common rule of thumb is that both np ≥ 5 and n(1 − p) ≥ 5 (some textbooks use 10). When p is near 0 or 1, or n is small, the binomial is skewed and no continuity correction will rescue the fit; use the exact binomial distribution instead. The continuity correction also applies to the Poisson-to-normal approximation and to normal approximations of other integer-valued distributions.