Log-Normal Distribution: Definition, Uses, and How To Calculate

Definition of Log-Normal Distribution

A log-normal distribution is a statistical distribution of logarithmic values from a related normal distribution. A log-normal distribution can be translated to a normal distribution and vice versa using associated logarithmic calculations.

Understanding Normal and Lognormal

A normal distribution is a probability distribution of outcomes that is symmetrical or forms a bell curve. In a normal distribution 68% of the results fall within one standard deviation and 95% fall within two standard deviations.

While most people are familiar with a normal distribution, they may not be as familiar with log-normal distribution. A normal distribution can be converted to a log-normal distribution using logarithmic mathematics. That is primarily the basis as log-normal distributions can only come from a normally distributed set of random variables.

There can be a few reasons for using log-normal distributions in conjunction with normal distributions. In general, most log-normal distributions are the result of taking the natural log where the base is equal to e=2.718. However, the log-normal distribution can be scaled using a different base which affects the shape of the lognormal distribution.

Overall the log-normal distribution plots the log of random variables from a normal distribution curve. In general, the log is known as the exponent to which a base number must be raised in order to produce the random variable (x) that is found along a normally distributed curve.

Applications and Uses of Log-Normal Distribution in Finance

Normal distributions may present a few problems that log-normal distributions can solve. Mainly, normal distributions can allow for negative random variables while log-normal distributions include all positive variables.

One of the most common applications where log-normal distributions are used in finance is in the analysis of stock prices. The potential returns of a stock can be graphed in a normal distribution. The prices of the stock, however, can be graphed in a log-normal distribution. The log-normal distribution curve can therefore be used to help better identify the compound return that the stock can expect to achieve over a period of time.

Note that log-normal distributions are positively skewed with long right tails due to low mean values and high variances in the random variables.

Lognormal Distribution in Excel

Lognormal distribution can be done in Excel. It is found in the statistical functions as LOGNORM.DIST.

Excel defines it as the following:

LOGNORM.DIST (x,mean,standard_dev,cumulative)

Returns the lognormal distribution of x, where ln(x) is normally distributed with parameters mean and standard_dev.

To calculate LOGNORM.DIST in Excel you will need the following:

x = value at which to evaluate the function

Mean = the mean of ln(x)

Standard Deviation = the standard deviation of ln(x) which must be positive