The Logistic distribution looks similar to the Normal distribution but has a kurtosis of 4.2 compared to the Normal kurtosis of 3. Examples of the Logistic distribution are given below:
Comparison of Logistic (0,1) (blue line) with a Normal distribution (red full) with the same mean and standard deviation
The Logistic distribution is popular in demographic and economic modeling because it is similar to the Normal distribution but somewhat more peaked. It does not appear often in risk analysis modeling.
The cumulative function has also been used as a model for a "growth curve". Sometimes called the Sech-Squared distribution because its distribution function can be written in a form that includes a sech2. Its mathematical derivation is the limiting distribution as n approaches infinity of the standardized mid-range (average of the highest and lowest values) of a random sample of size n from an exponential-type distribution.