Mathematical models, Making approximations when modeling real data using the normal distribution

 Mathematical models are abstract representations of real-world phenomena that use mathematical language and symbols to describe and quantify the behavior of a system or process. A good mathematical model should be based on accurate and relevant data, incorporate the relevant variables and parameters that affect the system or process being studied, and be able to make predictions or simulate outcomes under different scenarios. Mathematical models can be used to test hypotheses, make predictions, optimize processes, and inform decision-making in a wide range of fields, including physics, biology, economics, engineering, and social sciences.

There are several reasons why researchers might make approximations when modeling real data using the normal distribution:

1. Convenience: The normal distribution is a well-known distribution with many properties that make it easy to work with. For example, it has a simple mathematical formula, and its parameters can be estimated easily from data.
2. Assumptions: Many statistical models, including the normal distribution, are based on certain assumptions about the data. For example, the normal distribution assumes that the data is continuous and that the mean and variance are the only important features of the data. While these assumptions may not always hold true in reality, they can still provide a good approximation of the data in many cases.
3. Interpretability: The normal distribution has a clear interpretation in terms of the mean and standard deviation, which can help researchers to communicate their findings to others.

However, there are situations when researchers should not use the normal distribution to model their data. For example, if the data is strongly skewed or has outliers, the normal distribution may not be appropriate. In such cases, researchers may need to use a different distribution, such as the lognormal distribution or the t-distribution, that can better capture the characteristics of the data. Another example is when the data is discrete or categorical, such as the number of people in a household or the type of flower in a field, in which case a discrete probability distribution such as the Poisson or binomial distribution may be more appropriate.

As an example, consider the distribution of incomes in a given population. While the normal distribution may provide a good approximation for many populations, it may not be appropriate for populations with a large number of extremely wealthy individuals, which can result in a highly skewed distribution. In such cases, researchers may need to use a different distribution, such as the lognormal distribution, to better model the data.