Description: The log-normal distribution is a probability distribution that applies to a random variable whose logarithm follows a normal distribution. This means that if we take the logarithm of the variable’s values, they are distributed in a way that fits the classic bell curve of Gauss. This distribution is particularly useful in situations where data is positive and can span several orders of magnitude, such as in the case of incomes, asset prices, or wait times. The main characteristics of the log-normal distribution include its skewness, as it has a longer tail to the right, implying a higher probability of observing extreme high values compared to low ones. Additionally, this distribution is used in various fields of study, such as economics, biology, and engineering, due to its ability to model phenomena that do not distribute uniformly. In summary, the log-normal distribution is a valuable statistical tool that allows researchers and analysts to better understand data that exhibit significant variations and rightward biases.
History: The log-normal distribution was introduced into the statistical realm in the mid-20th century, although its foundations are based on earlier work on the normal distribution, which was formalized by Carl Friedrich Gauss in the 19th century. As scientists began to study natural and economic phenomena, they realized that many of these data followed a log-normal distribution, leading to its adoption in various disciplines. In particular, the work of statisticians such as George E.P. Box and Gwilym M. Jenkins in the 1970s helped popularize its use in time series analysis and economic data modeling.
Uses: The log-normal distribution is used in a variety of fields, including economics, biology, and engineering. In economics, it is commonly applied to model the distribution of incomes and asset prices, as these values tend to be positive and can vary widely. In biology, it is used to describe the size of organisms or the concentration of chemical substances in samples. In engineering, it is useful for modeling product lifetimes and system failures, where data may exhibit significant variability.
Examples: A practical example of the log-normal distribution can be observed in the analysis of the incomes of a population. If data is collected on the annual incomes of a group of people and the logarithm of those incomes is taken, the results are likely to fit a normal distribution. Another example can be found in the study of stock prices in the financial market, where prices tend to vary across multiple orders of magnitude, making the log-normal distribution a suitable tool for their analysis.
