Yes, I ran into that earlier: I tried to say "promedio" in a speaking question and it heard "media." Looking up both words in the two Spanish dictionaries I know, they explained a few things but stuck in enough things about midpoints to throw me off.
MEDIA: Is defined as "average," though WordReference also confusingly describes it as "mid-way between" and both say it is also "midfield" in a sports context (as for other contexts, hopefully no one is confusing averages and socks anytime soon): http://www.wordreference.com/es/en/translation.asp?spen=media http://www.spanishdict.com/translate/media
PROMEDIO: Is also "average," though both dictionaries agree that it also means "midpoint" ONLY when talking about distances: http://www.wordreference.com/es/en/translation.asp?spen=promedio http://www.spanishdict.com/translate/promedio
However, it's when one looks up translations for the English word MEDIAN that things get really complicated: In both dictionaries "la mediana" is the noun and "mediano/a" is the adjective. BUT WordReference also gives "media" as a translation for median in the context of statistics, which is just... gaaah >_<
tl:dr "Median" has its own set of words in Spanish, el promedio is fine for "average," la media is also "average" but might also be used as "median" by a few people.
In statistics, mean is the sum of all the values divided by the number of values that were added together. Median is the value where 50% of the measurements are higher and 50% are lower. Usually in statistics average isn't used since it has a much looser definition ex. "the performance was average"
I tend not to use this word, as I find it confusing and feel more comfortable with the mathematical terms, but you're right:
The verb is also nice to know I think (at least I use it more than the noun anyway)
I make a mental connection between "mean" and "preMEDIO". It's not a strong connection, and it also relies knowing that "mean" is a technical term used by statisticians to describe what most people think of as 'the average' of a group of numbers. That's the best I could come up with. Not great, but it works for me.