No, those clearly refer to the mean not the median. People do not ask the question "Given the range of heights in the population, what is the middle value", they want to know the mean height of the population.

This is especially easy to see when people talk about "average income", since it's clearly skewed by there being a long tail on the high-income side which pushes up the average salary. There's a reason why income is generally bracketed instead, and split into quintiles.

Look more carefully at my wording - most of the time, when you hear a news report using wording like "the income of the average family..." (as opposed to "average income"), you're hearing a median value, not a mean. The newsreader probably doesn't know it, but this is precisely because the median is less skewed by that long tail you mention.