lognormal distribution
The lognormal distribution describes a positive quantity whose logarithm is normally distributed. In plain terms: it is what you get when many small effects multiply together rather than add. Where additive noise piles up into a symmetric bell, multiplicative effects pile up into a right-skewed curve that hugs zero on the left and stretches out into a long tail on the right — exactly the shape of so many real claim amounts, incomes, and stock prices.
If you take the natural log of a lognormal variable, you get an ordinary normal with some mean μ and standard deviation σ — and those two numbers are the lognormal's parameters. Because of the exponential link, its mean is e^(μ + σ^2/2), comfortably above its median e^μ, the gap being a signature of its skew. The distribution is always positive, peaks at a modest value, and trails off slowly, so it readily produces the occasional very large outcome that a normal would deem nearly impossible.
The lognormal is one of the most widely used severity distributions in insurance: claim sizes, especially in liability and property lines, are frequently lognormal because losses tend to compound multiplicatively (a fire that spreads, an injury whose costs cascade). It is also the standard model for asset prices and investment returns in financial mathematics, underpinning option pricing and the modeling of variable-annuity guarantees. Its right skew and fatter-than-normal tail make it far safer than a normal for capturing the chance of a big loss — though for the very heaviest tails the Pareto can be heavier still.
Liability claim sizes are modeled as lognormal: most settle for a few thousand dollars, but a small fraction reach hundreds of thousands. The mean sits well above the typical (median) claim — the long right tail pulls the average up.
Multiplicative effects produce the lognormal — and its mean sits above its median.
Do not confuse the lognormal's parameters with its own mean and standard deviation: μ and σ describe the variable's logarithm, not the variable itself, so plugging the raw average into formulas as μ is a common and costly slip.