I ran a session of the Boston Teachers Math Circle on the topic of averages. We talked about different types of "averages": the arithmetic mean, the mode, the median... We even talked about the standard deviation, and other measures of spread.
But the bulk of the session was spent on exploring the fact that different data sets can have, for example, the same mean and the same standard deviation. Does that mean they "look" the same? You can find here my slides and my teaser flyer for this session.