Normalapproximationer

Normalapproximationer is a coined term used to describe an approach, method, or practitioner that applies the normal approximation to discrete probability models. The core idea is to approximate the distribution of a discrete random variable by a normal distribution with the same mean and variance, using the central limit theorem as justification.

Common use cases include the binomial distribution, where X ~ Bin(n,p) is approximated by a normal distribution with mean np and variance npq (q = 1 − p). For the Poisson distribution with parameter lambda, a normal approximation X ~ Poisson(lambda) is appropriate when lambda is large, using a normal with mean and variance both equal to lambda. In practice, a continuity correction is often employed to improve accuracy, for example approximating P(a ≤ X ≤ b) by P(a − 0.5 ≤ Y ≤ b + 0.5) with Y ~ N(np, npq). Z-scores are then used to convert probabilities to the standard normal scale.

The normalapproximationer may assess when the approximation is suitable, quantify its expected accuracy, and compare it with alternatives such as exact probabilities or other asymptotic methods (e.g., Berry–Esseen bounds, Edgeworth expansions). Limitations include reduced accuracy for small sample sizes or extreme parameter values, pronounced skewness, or very small tail probabilities, where exact methods or alternative approximations are preferred.

History and context: Normal approximations stem from the central limit theorem and the de Moivre–Laplace development, forming a standard tool in statistical practice for simplifying calculations with large samples. The term normalapproximationer is not widely used in formal literature but appears in informal or educational discussions about applying these simplifications.