Lectures on Categorical Data Analysis pp 41-56 | Cite as

# Normal Approximations

## Abstract

For the case of sufficiently large sample sizes, many properties of the binomial and multinomial distributions may be well approximated by normal distributions. The theoretical framework for such approximations is convergence in distribution. It is implied by the central limit theorem that when the sample size goes to infinity, appropriately normed binomial and multinomial distributions converge to the normal distribution. If, however, in the case of binomial distributions, also *p* converges to zero, so that *np* remains constant, the binomial converges to a Poisson distribution. The most important use of normal approximations is a very useful method to obtain asymptotic variance and covariance formulas for functions of binomial or multinomial variables. This *δ*-method is widely used in estimation and testing. Asymptotic normality applies to sampling distributions and refers to the deviation of the observed probabilities from their respective expectations. It does not affect the fundamental difference between categorical and normal assumptions with respect to the population distribution.

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