Does "x per y" always refer to the arithmetic mean?

Question

I recognize that there are many types of averages. That said, does "x per y" (e.g. visits per day) always refer specifically to the arithmetic mean?

For some context, I saw this recently where someone claimed the cited figure was actually in reference to a median. While I think it might have been OK to use the median when declaring something as an average, it seems to me that it is clearly wrong to denote anything besides the arithmetic mean with "x per y." This is highlighted by the common usage of "x/y" to indicate "x per y." Am I correct to always assume usage of the arithmetic mean here?

EDIT: Apparently my original post was not entirely clear. I am specifically referring to situations where y is unit of time, so clearly "x per y" is a frequency or a rate. I would always interpret it as the (arithmetic) mean frequency or mean rate. However, I saw somebody else interpret it as the median frequency (the median of the y discrete time intervals).

Answer 1

x per y is a ratio. It's not a mean of any sort. But taking the mean of ratios is dangerous in many situations. For example if you travel 240 miles at 60 miles per hour (x per y) and 240 miles at 40 miles per hour, your average speed is not 50 miles per hour. The first trip takes you 4 hours the second trip takes you 6 hours; total time is 10 hours, total distance is 480 miles, average speed is 48 mph.

So, please explain what you are trying to do.

Answer 2

whuber indicates that this is a frequency, which is the mean number of events at an instant. It is also a proportion, which is another perspective on a mean. The math is the same.

Perhaps the someone was referring to the median frequency, which could also be termed the median proportion or, more confusingly, the median mean.

An example: You measure each day the proportion of restaurant patrons who are female. This gives you a proportion/mean for each day. Perhaps you are interested in what the proportion is for a typical day. Maybe you do the median of this proportion rather than the mean.

In the example above, you arbitrarily chose median. If you are doing non-parametric statistics, then you would need to use the median.