The possibility should be considered that the underlying distribution of the data is not approximately normal, having "fat tails".

The distribution of need not be approximately normal (in fact, it can be uniform).

The distribution of the log odds ratio is approximately normal with:

He was experiencing an approximately normal weight, but he had seen no evidence of any mechanism whereby such an effect could have been achieved.

The affected leg usually has one foot that is in an approximately normal position.

If the two groups have the same survival function, the logrank statistic is approximately standard normal.

The distribution of the log Relative Risk is approximately normal with:

When he felt approximately normal again, he allowed himself to approach the road.

Thereafter it slowly diminished, until by evening they were able to make camp in an approximately normal forest.

Approximately normal distributions occur in many situations, as explained by the central limit theorem.