Lognormal Moments and Scale

The lognormal distribution models a random variable X whose logarithm is normally distributed. If ln X ~ Normal(μ, σ²), then X only takes positive values. The parameters μ and σ are the mean and standard deviation of the variable's natural logarithm, not of X itself.

Experiment

1. Set Log mean to 0 and Log standard deviation to 1. Observe that the mean (1.648...) is greater than the median (1.0), which is greater than the mode (0.367...).
2. Change Probe x to 1. The CDF is exactly 0.5, since the median is 1.
3. Set Log standard deviation to 0. The distribution becomes a Point mass at exp(Log mean). The density becomes unavailable, and the CDF becomes a step function.
4. Set Probe x to 0.5 when Log standard deviation is 0. Observe the CDF is 0. Change Probe x to 1, and the CDF jumps to 1.
5. Change Log mean to -1 and Log standard deviation to 1.5. Notice how the positive skew becomes even more pronounced.

Explore the interactive module here: https://riano.app/module/lognormal-moments