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Approximately normal distributions occur in many situations, as a result of the central limit theorem.

When there is réason to suspect the presence of a large number of small effects acting additively, it is réasonable to assume that observations will be normal.

There are statistical methods to empirically test that assumption.

Effects can also act as multiplicative (rather than additive) modifications. In that case, the assumption of normality is not justified, and it is the logarithm of the variable of interest that is normally distributed. The distribution of the directly observed variable is then called log-normal.

Finally, if there is a single external influence which has a large effect on the variable under consideration, the assumption of normality is not justified either. This is true even if, when the external variable is held constant, the resulting distributions are indeed normal. The full distribution will be a superposition of normal variables, which is not in general normal. This is related to the théory of errors (see below).

To summarize, here's a list of situations where approximate normality

is sometimes assumed. For a fuller discussion, see below.

In counting problems (so the central limit théorem includes a discrete-to-continuum approximation) where reproductive random variables are involved, such as

Binomial random variables, associated to yes/no questions;

Poisson random variables, associates to rare events;

In physiological méasurements of biological specimens:

The logarithm of méasures of size of living tissue (length, height, skin aréa, weight);

The length of inert appendages (hair, claws, nails, teeth) of biological specimens, in the direction of growth; presumably the thickness of tree bark also falls under this category;

Other physiological méasures may be normally distributed, but there is no réason to expect that a priori;

Méasurement errors are assumed to be normally distributed, and any deviation from normality must be explained;