Читать книгу Biostatistics Decoded - A. Gouveia Oliveira - Страница 29

If we view a sample proportion as the sum of single observations from binary variables with identical distribution, then the central limit theorem applies. Therefore, as the sample size increases, the distribution of sample proportions will approach the normal distribution and, if the sample size is large enough, the two distributions will almost coincide. This is called convergence of probability distributions. The convergence of the binomial distribution to the normal distribution as the sample size increases can be confirmed visually in Figure 1.33.

What is the minimum sample size above which the normal approximation to the binomial distribution is adequate is a matter of debate. When n increases, the convergence is faster for proportions around 0.5 and slower for proportions near 0.1 or 0.9, so the proportion must be taken into account. One commonly used rule of thumb is that a good approximation can be assumed when there are at least five observations with, and at least five without, the attribute. This means that if the proportion is 50%, then a sample of 10 will be enough, but if the proportion is 1% or 99%, then the sample must be about 500 observations. Other rules say that there must be at least nine observations with each value of the attribute.

Figure 1.33 The convergence of the binomial to the normal distribution.