Answer to Question #168883 in Statistics and Probability for GEORGETTE IZAC

Question

Mathematical models are used as tools to describe reality. These models are supposed to characterize the important features of the analyzed phenomena and provide insight. The normal distribution is an example of a random variable that is widely used by researchers to model real data.

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.

When forming your answer to this question you may give an example of a situation from your own field of interest for which a random variable can serve as a model.

Answer 1

Normal distribution assumptions are important to note because so many experiments rely on assuming a distribution to be normal. In most cases, the assumption of normality is a reasonable one to make.

Normal distribution assumptions can be relaxed in some situations but it forms a more complex analysis. If the physical process can be approximated by a normal distribution, it will yield the simplest analysis. However, some basic properties are retained even when distributions are not normal. For example, one might assume symmetry, as in a t-distribution even if the distribution is not truly normal.

In fact, a number of different non-normal distributions are just variations of the normal distribution. For example, a distribution might have a longer tail, which is a variation of the normal distribution. Such distributions too are frequently encountered.

The reason for the normal distribution assumptions is that this is usually the simplest mathematical model that can be used. In addition, it is surprisingly ubiquitous and it occurs in most natural and social phenomena. This is why the assumption of normality is usually a good first approximation.

However, there are important special scenarios when this is not the case. An understanding of the normal distribution assumptions will help researchers know the limitations of their experiment and also help them understand their own study and where it breaks down.