Log-Normal Distribution

What is a Log-Normal Distribution?

A Log-Normal Distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. In essence, the log-normal distribution is a plot of the logarithm of a normal distribution of a random variable. Unlike a standard normal distribution, the log-normal distribution does not follow a bell-curve, and is not symmetrical. Often, the distribution takes the shape of a normal distribution with a significant positive skew in one direction.


How does a Log-Normal Distribution work?

Log-Normal Distributions are often used in conjunction with normal distributions, as the two are related through logarithmic mathematics. The normal distribution can be converted to the log-normal distribution by taking the natural log of the random variables, in which the base is equal to e = 2.718. The log is the value of the exponent to which a number must be raised in order to produce the random variable that is along the normally distributed curve. Because log-normal distributions appear as the result of the logarithm of a normal distribution, the values contained in the distribution are always positive.


Applications of Log-Normal Distributions

As referenced above, the log-normal distributions contain exclusively positive values, unlike a normal distribution, which may contain negative values. This differentiator can prove valuable for those looking to analyze data using various distributions. For example, analysis of stock prices often turn to a log-normal distribution to display stock prices. While the potential returns of a stock may be plotted along a normal distribution, as it may include negative values, the log-normal distribution can better identify the compound return that a stock may achieve over time.