## 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 distributions 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.

For more, see also Investopedia's entry, *Lognormal and Normal Distribution*

## 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