Why is the Gaussian Distribution Called the ‘Normal’ Distribution?

The Gaussian distribution, commonly known as the normal distribution, is a fundamental concept in statistics. But have you ever wondered why it is called the 'normal' distribution? This article will explore the historical context, contributions of key figures, and the reasons behind its widespread use in various fields.

Historical Context

The term 'normal,' as used in relation to the Gaussian distribution, has its roots in the work of Francis Galton. In his book Natural Inheritance (1889), Galton referred to this distribution as 'the normal curve of distributions' or simply 'the normal curve.' He used the word 'normal' to mean 'usual' or 'what one typically finds in the situation.'

Central Limit Theorem

The Central Limit Theorem (CLT) is a key reason why the Gaussian distribution is so ubiquitous. According to the CLT, the sum or average of a large number of independent and identically distributed random variables tends toward a Gaussian distribution, irrespective of the original distribution of the variables. This theorem makes the normal distribution a common result in many natural phenomena, leading to its widespread adoption in statistical theory and practice.

Symmetry and Simplicity

The normal distribution is symmetric and characterized by its mean and standard deviation. This simplicity, combined with the fact that many real-world data sets tend to cluster around a central value, contributes to its classification as a normal distribution. The symmetry and simplicity of the normal distribution make it a convenient model for many data sets in various fields, including psychology, biology, economics, and more.

Widespread Occurrence

Many statistical methods and inferential techniques assume normality in data, making the normal distribution a standard reference point in statistics. This prevalence in various fields, such as psychology, biology, and economics, reinforces its status as a central distribution in statistical analysis. However, it is important to note that the normal distribution is not the only distribution used in these fields. Many human physical or social traits are normally distributed, but not all traits follow a normal distribution.

Alternative Name: Gaussian Distribution

It's worth noting that the Gaussian distribution also has an alternative name, which is the Gaussian distribution. This nomenclature is commonly used in physics texts, such as in statistical mechanics. The term 'Gaussian' is a nod to the mathematician Carl Friedrich Gauss, who contributed significantly to the early work in this area of mathematics.

Mixing of Meanings

One reason why the normal distribution is called the Gaussian distribution is that the word 'normal' in this context does not align with the common definition of 'usual' or 'typical.' In mathematics, 'normal' often refers to perpendicularity, which may be related to the symmetry of the distribution. This can lead to confusion, as many people associate 'normal' with 'expected,' 'common,' or 'most likely.' However, it is not surprising that many people think of it this way, as the distribution does come up frequently in various contexts.

It is also important to recognize that not all data follows a normal distribution. Many human physical or social traits are normally distributed, but not all traits follow a normal distribution. For example, power laws are non-normal distributions that are perfectly 'normal' in the statistical sense. This highlights the importance of understanding the limitations of the normal distribution and considering other distributions when necessary.

Conclusion

The term 'normal' in relation to the Gaussian distribution reflects both historical usage and the distribution's central role in statistical theory and practice. While the name 'Gaussian' is also used, both terms are recognized and understood in different contexts. The choice of terminology should be based on the specific field and audience, and changing one term to the other would cause unnecessary confusion.