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DA6. Invalidate Normal Distribution Approximation

Researchers often model real observations using the normal distribution, but sometimes the real distribution is a bit different from the perfect, normal distribution. List some reasons why researchers might make approximations like this and describe at least one situation when researchers should not use this approximation.

Modelling real-life problems can be complex, and these problems can take any shape or detail. The need to have a standard way to model different problems so that the results can be compared alongside the flexibility in reading the results and the need to simplify the calculations have allowed such approximations to appear.

According to Yakir, the normal distribution is continuous, while the binomial distribution is discrete, and the probability density of the normal distribution of number x scatters on the domain (x-1, x]. To use the normal distribution to approximate the binomial distribution, we have to calculate the probability of the value throughout the domain [x - 0.5, x + 0.5] (Yakir, 2011).

The Central Limit Theorem also indicates that the computation of a probability for a Binomial random variable is replaced by the computation of probability for a Normal random variable with the same expectation and standard deviation as the Binomial random variable(Yakir, 2011).

According to Jbstatistics, the binomial distribution is perfectly symmetric if p=0,5. However, it shows some skewness for any other value of p. The higher the distance of p from 0.5, the higher the skewness will be, and the more challenging it is to approximate the model correctly.

The normal approximation should be avoided if the p (probability of success ) is very low (close to 0) or very high (close to 1) unless the sample size is very large (Jbstatistics, 2012).

The normal approximation should also be avoided if the sample size is very small (Jbstatistics, 2012).

References

  • Jbstatistics, 2012. The Normal Approximation to the Binomial Distribution. https://www.youtube.com/watch?v=CCqWkJ_pqNU
  • Yakir, B. (2011). Introduction to statistical thinking (with R, without calculus). The Hebrew University of Jerusalem, Department of Statistics.