Lognormal Distribution Calculator

A positive random variable follows a lognormal distribution when its natural logarithm follows a normal distribution. If ln(X) is normal with location mu and standard deviation sigma, then X is lognormal with those parameters. The distribution is defined only for positive values and is usually right-skewed: many observations cluster at lower values while a long upper tail allows occasional large outcomes. The parameters describe the logarithms rather than the original measurements. Mu is the mean of ln(X), so changing mu shifts the distribution multiplicatively on the original scale. Sigma is the standard deviation of ln(X) and must be positive. Larger sigma values create greater relative spread, stronger right skew, and a heavier upper tail. Sigma is sometimes called the shape or log-scale parameter, not the ordinary standard deviation of X. The probability density function, or PDF, gives the relative density at a positive x value. Density is not itself a probability at a single point; probabilities come from areas over intervals. The cumulative distribution function, or CDF, gives the probability that X is less than or equal to the entered x. It is obtained by converting ln(x) to a normal z score and evaluating the standard normal CDF. On the original scale, the median is exp(mu), because half of the logarithms lie below mu. The mode is exp(mu minus sigma squared), which lies below the median for positive sigma. The mean is exp(mu plus half sigma squared), which lies above the median because large upper-tail values pull the arithmetic average upward. Their ordering, mode below median below mean, is a characteristic feature of right skew. The variance equals the quantity exp(sigma squared) minus one, multiplied by exp(2mu plus sigma squared). It can grow rapidly as sigma increases. This sensitivity is important in financial, environmental, reliability, biological, and duration data, where a few large observations may dominate totals. Report parameter definitions clearly because some software uses alternative parameterizations based on the original-scale mean or a separate scale factor. A lognormal model is plausible when values are strictly positive and multiplicative influences combine to produce the outcome. Examples can include incomes, particle sizes, repair times, biological concentrations, and some asset-price ratios. It is not automatically suitable for every positive skewed sample. Check a histogram, a log-scale plot, and a normal probability plot of the logarithms. Zeros and negative values cannot be represented without a scientifically justified transformation or a different model.