Description: The log-normal distribution is a probability distribution that describes a random variable whose logarithm follows a normal distribution. This means that if a random variable X has a log-normal distribution, then its natural logarithm, ln(X), is normally distributed. This characteristic makes the log-normal distribution particularly useful for modeling phenomena where values are always positive and can vary in orders of magnitude, such as incomes, asset prices, and particle sizes. The shape of the log-normal distribution is asymmetric, with a longer tail to the right, indicating a higher probability of extreme high values compared to low ones. This asymmetry is one of its most distinctive features and the fundamental difference from the normal distribution, which is symmetric. The log-normal distribution is defined by two parameters: the mean and the standard deviation of the logarithm of the variable. Its relevance in applied statistics lies in its ability to model data that do not fit well to normal distributions, providing a valuable tool for data analysis across various disciplines, including economics, biology, and social sciences.
History: The log-normal distribution was first introduced in the context of statistics by American mathematician George W. Snedecor in 1937. However, its use became popular in the 1960s when it was applied in various fields such as economics and biology. Over the years, several researchers have contributed to its understanding and application, highlighting its importance in the analysis of data that exhibit positive skewness.
Uses: The log-normal distribution is used in various disciplines, including economics, where it is applied to model income distribution and asset prices. It is also common in biology to describe the size of organisms or particles. In engineering, it is used to model material strength and in environmental studies to analyze pollutant concentrations.
Examples: A practical example of the log-normal distribution is the analysis of income within a population, where most individuals earn relatively low amounts, but there are a small number of individuals with extremely high incomes. Another example can be found in the distribution of stock prices in the financial market, where prices can vary significantly and do not follow a normal distribution.
