Continuity Correction

Probability · Axiom Academy

Improving normal approximations for discrete distributions 1. Why We Need Continuity Correction Consider X ~ B(20, 0.5). The exact probability P(X = 10) uses the discrete PMF. But when we approximate with the normal distribution: This is wrong! The normal distribution is continuous, so the probability at any single point is zero. The solution: Treat the discrete value 10 as representing the interval [9.5, 10.5]. Then: 2. Continuity Correction Rules When approximating discrete X with continuous normal, apply these corrections: P(X = k) → P(k - 0.5 < X < k + 0.5) The pattern: add 0.5 when moving away from the mean, subtract 0.5 when moving toward the mean. 3. Example: With vs. Without Correction Problem: Let X ~ B(100, 0.3). Find P(X ≤ 25). Without continuity correction: Computing: z = (25 - 30)/4.58 = -1.09, so P(Z ≤ -1.09) ≈ 0.138 Computing: z = (25.5 - 30)/4.58 = -0.98, so P(Z ≤ -0.98) ≈ 0.164 Exact binomial: P(X ≤ 25) = 0.165 The continuity correction (0.164) is much closer to the exact value (0.165) than the uncorrected approximation (0.138)!

This is the written version of the interactive lesson above. See the full Probability course.