For a short-tailed distribution, the tails approach zero very fast. Such distributions commonoly have a truncated ("sawed-off") look. The classical short-tailed distribution is the uniform (rectangular) distribution in which the probability is constant over a given range and then drops to zero everywhere else--we would speak of this as having no tails, or extremely short tails.

For a moderate-tailed distribution, the tails dive to zero in a moderate fashion. The classical moderate-tailed distribution is the normal (Gaussian) distribution.

For a long-tailed distribution, the tails dive to zero very slowly--and hence one is apt to see probability a long way from the body of the distribution. The classical long-tailed distribution is the Cauchy distribution.

In terms of tail-length, the histogram shown above would be characteristic of a "short-tailed" distribution.

The optimal (unbiased and most precise) estimator for location for the center of a distribution is heavily dependent on the tail-length of the distibution. The common choice of taking N observations and using the calculated sample mean as the best estimate for the center of the distribution is a good choice for the normal distribution (moderate-tailed), a poor choice for the uniform distribution (short-tailed), and a horrible choice for the Cauchy distribution (long-tailed). Although for the normal distribution, the sample mean is as precise an estimator as we can get, for the uniform and Cauchy distributions, the sample mean is unduly noisy.

For the uniform distribution, the midrange

midrange = (smallest + largest) / 2