Description: The univariate normal distribution is a probability function that describes how the values of a single random variable are distributed around its mean. This distribution is characterized by its bell-shaped curve, symmetric around the mean, indicating that most data points cluster near this central value, while the probability of finding extreme values decreases as we move away from the mean. The normal distribution is fully defined by two parameters: the mean (μ), which indicates the center of the distribution, and the standard deviation (σ), which measures the spread of the data. Mathematically, the probability density function of a univariate normal distribution is expressed as: f(x) = (1 / (σ√(2π))) * e^(-((x – μ)² / (2σ²))). This formula shows how the probability of a specific value is determined based on its distance from the mean, normalized by the standard deviation. The normal distribution is fundamental in statistics and is used as a model for natural and social phenomena, as many processes tend to follow this distribution under certain conditions. Its importance lies in the central limit theorem, which states that the sum of a sufficiently large number of independent random variables tends to be normally distributed, regardless of the original distribution of the variables.
History: The normal distribution was introduced by French mathematician Pierre-Simon Laplace in the 18th century, although its modern form was developed by Carl Friedrich Gauss in 1809, who used it to describe measurement errors in various fields. Throughout the 19th century, the normal distribution became established as a fundamental statistical tool, especially in the field of statistical inference and error theory.
Uses: The normal distribution is used in various fields, such as statistics, psychology, economics, and engineering. It is fundamental for making statistical inferences, such as hypothesis testing and constructing confidence intervals. Additionally, it is applied in quality analysis and process control, as well as in modeling various phenomena in natural and social contexts.
Examples: A practical example of the univariate normal distribution is the height of an adult population, which tends to follow a normal distribution with specific mean and standard deviation. Another case is the performance on a standardized test, where most students score close to the mean, with fewer students at the extremes.
