Character mistake: When the posse is heading out in the morning, one guy says that it's heading north and they start riding in the direction he's pointing. However it's supposed to be first thing in the morning, and you can see the shadows of the horses cast to the left of screen while they are riding towards the camera. The sun rises in the east, which would be right of screen with the shadows cast towards the west. Since they are riding towards camera, this puts them heading south.
Character mistake: Nicolas Cage looks up the Bing results for Alan Marsh, the journalist he accidentally kind of murdered. The URL of his personal blog, hosted by the newspaper he works for, spells "blog" wrong. (00:55:25)
Character mistake: In the pivotal scene where Bolly and Jonah are explaining the need to replace their departing players in the aggregate, Jonah adds up 3 players on base percentage and divides by three. You can't get an accurate average for a rate stat by that means unless all the players had the same number of plate appearances, which they did not. The answer given by Jonah Hill is 14 points low. When the older scout questions it, Beane scolds him that he "didn't carry the one." Same mistake is in the book.
Character mistake: Near the end of the movie, Dr. Bennett told Nathan that he could live with her and be able to finish the school year (without having to transfer). Assuming Nathan was a minor (under 18 years of age), Dr. Bennett did not have the legal authority to simply take custody of Nathan. There are proper court procedures to follow; a judge would have to agree and sign a court order declaring Dr. Bennett as Nathan's (temporary or permanent) legal guardian.
Character mistake: When travelling between zones, they go 12 - 10 - 9 - 8 - 7 - 5 - 4. Which is the only route you can take that includes zone 8 as shown. This is silly as there are quicker routes such as 12 - 7 - 5 - 4.