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Not that this is something you can do yourself, but... When I used to
work tech support, we had to keep track of our calls, how long each
one took, what the reason for the call was, etc. This was all tracked
on a database. One of the items we could check was RTFM. If your tech
support group has to keep these kinds of records, it might not be
difficult to add that one item, especially if it is just a mouse click
sometime during the call.
My company provides free phone support. Our customer base is continuously
growing. Our support staff is stretched pretty thin. I can think of no easy
way to track which tech calls could have and could not have been resolved by
reading the book. Similarly, I can think of no way to measure a relative
increase or decrease in support calls resulting from a change in
documentation, and isolating other variables, like more intuitive GUI.