Why Gaussian Distribution is so fundamental to Statistics?

While learning statistics you often start with Gaussian or Normal distribution. I wondered why it is basis of so many models and statistical assumptions.

The Gaussian distribution, (also known as the Normal distribution) is a probability distribution. Its a bell-shaped curve is dependent on?the mean and?the standard deviation.

It is given by this Probability density function (PDF):

where:

• x?= value of the variable or data being examined and f(x) the probability function
• μ = the mean
• σ = the standard deviation

Let us go over some key reasons:

1. Central Limit Theorem (CLT): One of the most critical reasons is the Central Limit Theorem. This theorem states that, under certain conditions, the sum of a large number of random variables, irrespective of their distributions, will tend to follow a Gaussian distribution. This makes the Gaussian distribution a natural choice for modeling a wide range of phenomena, as many real-world processes can be thought of as the sum of several small, independent effects.
2. Mathematical Properties: The Gaussian distribution has convenient mathematical properties. It is symmetric and defined by just two parameters - the mean and the standard deviation, which makes it relatively simple to analyze and interpret. Its shape is entirely determined by these two parameters, making it easy to estimate from data.
3. Ubiquity in Natural and Social Phenomena: Many natural and social phenomena approximately follow a Gaussian distribution, particularly those that reflect aggregated effects of many small, random disturbances. Examples include heights of people, measurement errors, blood pressure readings, and many other biological measurements.
4. Linear Models: Gaussian distributions are the foundation of linear models. The assumption of normally distributed errors is central in ordinary least squares regression, a fundamental technique for statistical modeling and inference. This assumption simplifies the mathematics involved in estimation and hypothesis testing.
5. Generalization to Other Distributions: The Gaussian distribution serves as a basis for other important distributions. For example, the chi-squared, t-distribution, and F-distribution are all related to the normal distribution and are derived under specific circumstances.
6. Analytical Convenience: In many statistical methods and tests, assuming a Gaussian distribution simplifies the formulas and calculations. This often leads to exact analytical solutions to problems that would be much more complicated otherwise.
7. Historical Precedence: The Gaussian distribution has been used extensively since the 18th century. This long history means that a vast amount of statistical methodology and theory has been developed with the Gaussian distribution in mind.

It's important to note, however, that not all data are normally distributed, and there are many situations where other distributions are more appropriate. In such cases, different models and techniques are used. Nonetheless, the Gaussian distribution's simplicity, mathematical properties, and relevance to many real-world situations make it a cornerstone of statistical modeling and analysis.

A pretty popular empirical rule for Gaussian distribution:

Origin and Early Development

De Moivre's Initial Formulation (1733): The earliest version of the normal distribution was formulated by Abraham de Moivre in his work "The Doctrine of Chances" in 1733. De Moivre was trying to approximate binomial distributions, particularly for large values of the number of trials. He introduced a formula that is now recognized as the normal distribution's probability density function. Later Laplace's Further Developments (1770s-1780s) expanded de Moivre's findings. He used the normal curve to approximate the distribution of errors and made substantial contributions to the central limit theorem, which explains why the normal distribution arises so commonly.

Naming and Gauss's Contribution

Carl Friedrich Gauss (Late 18th Century): The name "Gaussian distribution" is primarily attributed to Carl Friedrich Gauss, a German mathematician (if you haven't read a story of little Gauss, go check it out!). Gauss used this distribution in 1809 to analyze astronomical data (refer to my previous post if you'd like to know history on this, a really interesting one!). He developed the method of least squares, which assumes that the errors in observations are normally distributed (check one of my past posts to know the history on this - I assure you its quite interesting!). Gauss's work in the theory of errors and the method of least squares significantly popularized the normal distribution in the scientific community -- but why "Gaussian?": The distribution was named "Gaussian" after Gauss, although Gauss was not the first to use or describe this distribution. This naming is somewhat a historical artifact, reflecting Gauss's immense influence and the widespread adoption of his methods.

Later Developments

Adolphe Quetelet (19th Century): Quetelet, a Belgian astronomer, mathematician, statistician, and sociologist, applied the normal distribution to social statistics and biological measurements, broadening its scope beyond astronomy and mathematics. Later Sir Francis Galton's (Late 19th Century) work in biometrics and the study of heredity further popularized the normal distribution. He used it to analyze a variety of biological data, laying the groundwork for its widespread application in the biological and social sciences.

Modern Usage

Today, the normal (or Gaussian) distribution is a cornerstone of statistical analysis, used in a wide range of fields from physical and social sciences to engineering and economics. Its mathematical simplicity and natural occurrence in many real-world scenarios make it one of the most important and ubiquitous concepts in statistics and probability theory. Here are some examples of its usage across different disciplines:

1. Statistics and Data Analysis:Hypothesis Testing: In statistics, the normal distribution is used for hypothesis testing, especially in t-tests and z-tests. These tests often assume that the data or the sample means are normally distributed.
2. Confidence Interval Estimation: When estimating confidence intervals for means and variances, the normal distribution is a standard assumption, particularly for large samples.
3. Finance and Economics:Risk Assessment and Modeling: In finance, the Gaussian distribution is used to model asset returns and assess risks. It's a foundational element of the Black-Scholes model for option pricing.
4. Economic Modeling: Economists use the normal distribution to model a variety of economic phenomena, such as consumer behavior, income distribution, and market returns.
5. Science and Engineering:Error Analysis: The normal distribution models measurement errors and uncertainties in scientific experiments and engineering applications.
6. Quality Control: In manufacturing and engineering, it's used for quality control and reliability analysis. For example, the six sigma methodology assumes normally distributed errors.
7. Social Sciences:Behavioral and Social Research: In psychology and sociology, the normal distribution assists in analyzing and interpreting data related to human behavior and societal trends.
8. Educational Assessment: It's used in educational testing and assessment to interpret scores and grades, assuming that student performance follows a normal distribution.
9. Medicine and Public Health:Clinical Trials: In medical research, the Gaussian distribution is used for the design and analysis of clinical trials, especially in the evaluation of drug effects.
10. Epidemiological Studies: It helps in the analysis of epidemiological data, such as the distribution of health-related variables in a population.
11. Environmental Science:Pollution and Environmental Modeling: In environmental science, it's used to model pollution dispersion, environmental risks, and to analyze environmental data.
12. Machine Learning and Artificial Intelligence:Data Normalization: Gaussian distributions are used for normalizing data in machine learning algorithms to improve performance.
13. Probabilistic Models: In AI, Gaussian models underpin various probabilistic models, such as Gaussian mixture models used in clustering and classification tasks.

These applications illustrate the versatility and fundamental importance of the Gaussian distribution across numerous fields. Its ability to model a wide range of natural and human-made phenomena makes it an invaluable tool in research and practical applications.

Here is an excellent follow-up article on Gaussian distribution. Happy learning!