Lognormal Distribution Calculator

A continuous random variable X follows a lognormal distribution if its natural logarithm, Y = ln(X), is normally distributed with mean mu and standard deviation sigma. Consequently, the parameters mu and sigma represent the mean and standard deviation of the underlying normal distribution, rather than the lognormal variable itself. The probability density function (PDF) is defined as f(x) = (1 / (x * sigma * sqrt(2 * pi))) * exp(-(ln(x) - mu)^2 / (2 * sigma^2)) for x > 0. Key metrics like the mean, variance, median, and mode are derived directly from mu and sigma using specific exponential formulas: the median is equal to exp(mu), while the mean is calculated as exp(mu + (sigma^2) / 2).

Consider an analysis of real estate property values where the underlying log-transformed data yields a mean mu = 0.5 and a standard deviation sigma = 0.2. To find the median property value, we compute the exponential of mu: median = exp(0.5) = approximately 1.6487. To find the expected mean value of the distribution, we apply the formula exp(mu + (sigma^2) / 2), which gives exp(0.5 + (0.2^2) / 2) = exp(0.52) = approximately 1.6820. If we want to evaluate the probability that a value falls below x = 1.5, we evaluate the cumulative distribution function (CDF) using these parameters, yielding a specific probability percentile for the dataset.

Always verify whether your input parameters describe the lognormal variable directly or its underlying normal parameters (mu and sigma). Remember that lognormal variables must always be strictly greater than zero; trying to input zero or negative numbers will result in undefined mathematical states. When analyzing heavily right-skewed data like income distributions or response times, compare both the mean and the median to get an accurate representation of central tendency, as outliers can heavily inflate the mean.