The Tigers finished the month of June with a record of 15-13 but it could have been a lot worse. By any other measure other than victories, the Tigers had a poor month. They were outscored by opponents 134-112 which means they were 9th in the league runs scored and second in runs allowed. By the Pythagorean Theorem, a team with those numbers would typically go 11-17, not 15-13.
They were outhit and/or pitched by a wide margin in most categories: .273 to .249 in batting average, .365 to .319 in on base percentage, .439 to .417 in slugging average, .804 to .736 in OPS, 126 to 86 in walks, etc. Those are not the numbers of a winning team. They won because they were able to win a lot of close games: an 11-4 record in games decided by two runs or fewer.
You can look at things a a couple of different ways. You could say that they were lucky to win as much as they did despite being outplayed statistically. Or you might suggest that they have a talent for performing in the clutch and winning close games. Personally, I side more with the former than the latter but they did win those games and the wins can't be taken away. I see it as them going through a period where they played poorly but still managed to win and to keep their division lead.
Whatever way you look at it, they need to start playing better if they want to keep their lead. The next series will be a big one. They will be facing the Twins in the dreaded Metrodome, a place where usually only the Twins get lucky.