When I am playing Hearthstone, I sometimes get little messages about what my friends are doing in the game, such as:
I am confused about why it sometimes tells me my friend's Arena record when they finish and sometimes just tells me that they have finished. Is there some pattern to when it tells me Arena records of my friends? Does it only notify friends if the record is better than 4 wins?
I added a comment to your question, just as a joke, but after research it is in fact the right answer.
When someone from your friendlist is finishing arena with a score of 3 or less you get the " just finished an Arena run." message, but when it is 4 or more, it notifies you their score too, so you were right !
The sources I could find where forums as it appears to be like that since the notification was introduced, and the patchnote isn't precise :
Social – When your friends complete an Arena run or get a Legendary card in a pack, you’ll be notified so you can tell them, “Grats!”
