I’m sure it’s happened before, and I know it will happen again. Just as I have so often, you’re going to be wrong. Now what?
Well, first is a big don’t. Don’t act like it didn’t happen. Don’t hide it or cover it up. Your team will figure it out; in fact, they’re likely to know before you do. So they’re watching to see how you handle it.
The good news is, you don’t necessarily have to apologize. Often just correcting yourself is enough. “Hey, guys, I said this, but when I dug into it, turns out that’s not right. Here’s what I found out.” That kind of thing builds trust, and gives them permission to correct themselves too.
If you argued with someone about it, or you caused someone extra work, then you do need to apologize. “I shouldn’t have argued with so-and-so. She was right, and I was wrong. I’m sorry.” Or, “I blew it and made you guys do a bunch of unnecessary work. I’m sorry.” You can’t undo it, but you can own up to it, which at least lets everyone know you care.
Better still is if you can tell them something you’ve done to make sure you won’t make a similar mistake next time.