Sam Savage has an excellent book by the title “Flaw of Averages” and also does a great talk on it.  An example often used is that someone can easily drown in a lake that has an average depth of 4 feet.  Of course it may only be a few inches in some places, but 30+ feet in others, and it is the latter where people are more likely to drown.

An example I like is average lifespan.  We keep hearing that people are living longer (on average), but in fact the lifespan of an individual hasn’t really changed significantly in decades.  The average, however, has improved since far fewer infants die at birth or shortly afterwards.

It’s dangerous to use average of you don’t also know the variance, and how they apply to the particular situation under analysis.