### DEFINITION of Log-Normal Distribution

Log-normal distribution is a statistical distribution of random variables that have a normally distributed logarithm. Log-normal distributions can model a random variable *X*, where log(*X*) is normally distributed.

These distributions, under multiplication and division, are self-replicating. That is to say, multiplying or dividing log-normal random variables will result in log-normal distributions.

### BREAKING DOWN Log-Normal Distribution

For example, log-normal distributions can model certain instances, such as the change in price distribution of a stock or commodity positions. This is because the time series creates random variables. By taking the natural log of each of the random variables, the resulting set of numbers will be log-normally distributed. Other examples of suitable applications of log-normal distribution analysis include survival rates of cancer patients and failure rates in product tests.

Log-normal distributions are positively skewed due to low mean values and high variances in the random variables.

See Investopedia's entry, "Lognormal and Normal Distribution," to learn the main differences between the two types.