In what way might an average be misused?

When you were a child, you might have seen a dark brown bottle of a mysterious liquid high on a shelf in your garage, out of your reach. “That’s poison,” a parent might have warned, pointing to the container. “Don’t touch.” It nonetheless had some use. It was there for a reason. This example illustrates a principle: Items that can be misused often have a valid use, and vice versa. An average (e.g. mean, median, and mode) is not an exception; it can be useful, but if it is misused or misinterpreted it can be destructive.
In what way might an average be misused? Alternatively, how might an average be misinterpreted? For instance, how can a misinterpreted average pertain to stereotyping? How can we avoid misuse or misinterpretation of averages? Provide a specific example to illustrate your explanation.