DiMaggio's average was .409 through Sept. 9th but he hit only .233 in his last 73 ABs to finish at .381. One report said he had a virus that had sidelined pitcher Red Ruffing and "Years later Joe would say the cold he had developed on the 9th had caused a painful eye twitch."
But manager Joe McCarthy would not give him a couple of games off to recuperate.
"Can we believe that McCarthy didn’t know that his greatest star — the brightest star in America — was in agony? If he did know, did he really make DiMag stay in the game and suffer? Why???!!! McCarthy was an alcoholic, who kept a brown bag with him on the end of the bench, where he sat alone while the coaches apparently ran the team. He was sometimes referred to as a “push button manager,” the closest anyone came to whispering the problem. Eleven years later he would be found literally in the gutter and was swiftly and quietly whisked home to Buffalo and oblivion. No one then, or now, uttered the word “alcoholism.”"If the coaches ran the team, it seems like they could have taken DiMaggio out of the lineup or after just one AB to get him some rest.
DiMaggio finished with 462 ABs. That means he needed 185 hits to reach .400. He was 17 for his last 73, and finished with a total of 176 hits. So he was 9 hits short of .400. If he had gone 26 for his last 73, he would have made it. That is still a .356 average and even a guy at .400 has no guarantee of reaching that level for a short spurt.
Using the binomial distribution function in Excel, I get that there is a 15% chance of DiMaggio getting 25 or fewer hits in his last 73 ABs (assuming he was truly a .409 hitter).
But was he truly a .409 hitter that year? Or could have been expected to be in Sept? Players usually hit worse in Sept., perhaps due to colder weather. The Yankees as a team batted .270 in Sept. while it was .287 for the whole season (it was .290 thru Aug.). Using the Baseball Reference Play Index, it looks like the Yankee regulars were still used as much as usual in Sept. Maybe with a big lead, they did not quite have their usual incentive. They never lead by less that 11.5 games the whole month. Anyway, if we reduce DiMaggio by 20 points then we expect only a .389 average from Sept. 10th on.
DiMaggio himself batted .319 in Sept/Oct games in his career while it was .325 overall (.326 thru Aug. for his career). So we had to expect some drop off. Also, in his last 19 games, only 2 were on the road and Yankee stadium was not a good place for righties like DiMaggio (it was 467 feet to left center). In his career, he batted .315 at home and .334 on the road. So maybe we reduce him .008 for this. He is down to .381
DiMaggio also had 78 of his 109 ABs in Sept. against righties. That is 71.6%. It was 66.7% for the whole season. Maybe that is 5 ABs more than usual against righties. In his career, DiMaggio batted .342 vs. lefties and .316 vs. righties. This would be a slight effect. Maybe .001. So he is down to .380
That would get him a .400 AVG for the season (and he finishes at .403 or .404). But what are the odds he can still reach .356 if we assume he was a true .380 hitter?). Using the binomial distribution again, I get a 30% chance that he gets 25 or fewer hits in his last 73 ABs if he is a true .380 hitter. That is a high enough chance that we can't assume it was just his illness, eye twitch and lack of rest due to a bad manager that cost him his .400 season.
Now I might have penalized him too much for a lower Sept. average by the Yankees as a whole. If I cut that penalty in half and we make him a true .390 hitter, there is still a 24% chance he does not get the 26 hits he needed in those last 19 games to finish at .400.