People use "mean" and "average" in way that suggest they're the same, but one certainly doesn't have to say "average (herein defined as the mean)" because that's the overwhelmingly dominant meaning anyway.
In fact, I'd say that "averagING" (as a verb) almost has to be the mean; I'd find it deceptive if it were mode-finding.
You're right that people often conflate the mean and median, but I think that's more to do with underestimating the very, very long tails of some distributions.
I certainly agree that people mix them up. If you ask someone to quickly estimate the average/mean of something, you are likely to get some other location measure, like the median or mode. Some of this might be a heuristic: it's easier to mentally estimate the middle/most common item than it is to keep a running sum, and for many distributions, you get a similar answer. Some of it might be due to fogginess over the definitions. And, of course, these other summary statistics are often closer to what people regard as a `typical' value, especially if there's a long tail, so it
On the other hand, suppose we had some data like this
X=[2, 2, 2, 4, 4, 8, 16]
If I asked you what the average of X was, the answer I'm expecting is "5.4", or perhaps "the median is 4", "the mode is 2", etc. I would be surprised if you just answered "2" or "4", and once I figured out what you did, I would find it annoyingly pedantic.
I've worked with a lot of different people from different backgrounds, and I don't think anyone has ever used (unadorned) average to mean something other than the mean. Perhaps your experience is different.
Alright, this is rooted in a confusion between the normal use in statistics and the normal use in human speech.
Absolutely, average-without-qualifications means the mean, if we're going to be handed a list of actual numbers and expected to produce another number from them.
But especially for highly right-tailed distributions, when people talk about the average, they mean something more like the median, if there's a significant difference.
I would find it annoyingly pedantic if someone said the average net worth in Jeff Bezos' municipality was 60mm, if the median was closer to 0.75mm. If I was handing someone an Excel spreadsheet of tax returns, I would of course tell them that we needed the median. Or the mean but in this circumstance I wouldn't use the word 'average' because of the wide skew. If I did anyway, yeah I would probably get the mean back.
That's how average works in statistics, but not in common speech.
Two random examples from other tabs I had open: https://direct.mit.edu/neco/article/16/3/477/6879/Mean-Insta... or https://www.cell.com/neuron/fulltext/S0896-6273(13)00798-8 Mean and average are clearly the same thing, but only via context.
In fact, I'd say that "averagING" (as a verb) almost has to be the mean; I'd find it deceptive if it were mode-finding.
You're right that people often conflate the mean and median, but I think that's more to do with underestimating the very, very long tails of some distributions.