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Do they mean different things, or pretty much the same?

For example:

I typed on Google "How many breaths does a person take in a day?" and scrolled down until I spotted an/the average: 17,000-30,000.

Is the meaning different if you pick an or the?

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Language reveals the speaker's thought. It is subjective, not objective.

I scrolled down [through the Google search results] and found the average.

If a speaker says the average the speaker considers this average to be in accord with an average computed in a perfect world where all breathers are taken into account, or with a sample size statistically so large and diverse that the effects of additional breathers would be de minimis given the required precision. Or perhaps the speaker is not even aware of such vagaries but is someone who takes at face value and treats as gospel whatever Google coughs up. In The speaker either believes the average to be definitive or does not even realize that such calculations can be flawed.

I scrolled down [through the Google search results] and found an average.

an implies that the speaker believes there are, or at least could be, other averages that might differ from this one. For whatever reason, the speaker is not going to confer upon this average the status "definitive". The speaker again might not even be aware of the sort of inaccuracies that creep into such calculations. All the speaker knows is that when you search on Google for such things, the results will often include one or more hits presenting averages, and this is one of them.

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An average person is synonymous with "a typical person". On the other hand "the average person" would mean the median (if taken literally).

Nevertheless, there is not much actual difference in non-technical language. In technical language, you should avoid "average" and specify "mean" or "median"

The mean number of breaths a person takes is about 15000.

The median person takes 14000 breaths every day.

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