Not sure why the picture didn't show up

Erick, I see you've returned. You'll get several suggestions just as you did on the other issue. I'm confidant all will be correct. Several ways to skin a cat. ( sorry pet lovers ) . No, probably nothing sinister. This is not much different then your wall issue. Look at the floor as you did your wall. I see your tile backer is not installed yet. You can rip a furring strip and install in the low spot. Prior to setting the tile backer you'll need to fill the void. Thin set / mortar / FixAll might even work. You don't want any air space between the subfloor and the tile backer. Some guys will just go ahead with the tile backer as is and slap extra thin set under the tiles. Your 3/8" in my opinion, is right at the tipping point for that technique.

Self leveling compound. But is the problem happening at the wall where your level is pointing? If that is the case, the wall landing on the joisting in that section is picking up incidental loading. Stabilizing the problem is a good idea. It's all about joist direction and roof or second floor load and gravity just plain sucks.

I would use a self leveling underlayment such as Level Quick. The cement backer board has to be set in thinset according to all cement backer board manufacturer's specifications.

Also, sometimes flat and straight are a better alternative to level. Take a very typical bathroom that is 5' wide x 10' deep and tub is on the far wall ( opposite the door ). It makes complete sense to start laying you tile at the tub and back your way out to the door, right?, but say the floor is level from side to side but drops 1" from the tub to the door. Suppose you have hardwood, carpet or laminate floor in the hallway anywhere from 3/8" thick to 3/4" thick. If you want the floor level, and taking into account the thickness of the tile backer, tile, and mortar ( about 3/4" total ) in best case scenario the finished tile surface is 1" above hallway floor treatment. Layout is everything

Excellent point Yarbro, don't give away all the secrets 8-)

Ironwood brings up a very good point. If the wall is running the same direction as the floor joists and sits between the joists you very well could get a 3/8" belly in the plywood subfloor. You say, " the joists actually drop that distance" What direction do the joists run in relationship to the wall in the photo?

Erick seems like a good guy Norm. He actually climbed into the dreaded crawlspace and investigated. I'm impressed

That's why he has all of us over here helping...no fear and a good mind.

When I see him in a picture with a rat skeleton on a leash I'll believe it. 8-)

Lol! Very funny Norm ;) And rat nuggets stuck in his ear canal

The wall is running with the joists. Going back to the crawl I have discovered that the joist deflects that much between the foundation and the Beam. In plane at the foundation/beam and sagging in the middle.

Then if its just a few joists and thats the issue than just get a jack raise it up 3/8 and put some metal shims between the foundation wall and the joists and then let it back down.

Then I will be high at the perimeter wall, no?

no if you jacking up where they have sunk your just putting them back to their original place and the wall will go with it. you dont even need to go all the way even if you split the difference and got it ti within 3/16 you could use mortar to float that out.

could I just use a jack post to do the raising?

ya but it will be easier with a pump jack you can get one at northern tool for like 50 bucks

Also, if my joist is deformed due to deflection is it going to go back to being level if I jack it up?

If the joist in question has bowed in the middle and it is under a wall carrying load from upper story or roof, if possible I would recommend bringing in an LVL joist to sister to the joist in question. You can use the new LVL to jack up floor to level, glue and screw it to exisitng floor joist and if at all possible get it to rest on support wall at each end (not easy but possible), if not use a screw jack or other permanent support to help carry the load to the foundation.

To my knowledge, the joist is carrying no vertical load. The only wall on the runs parallel to the joists and is a 4'-0" wide (obviously floor to ceiling) stud wall. My theory is that it has deflected due to the weight of the bathtub filled with water.

I doubt that the bathtub filled with water would cause the deflection as it would not be a constant load (if it was a heavy cast iron tub then perhaps, but probably just one of those things that wood is prone to doing, especially after 50 years), but it doesn't really matter at this point why the joist bowed. Sistering a joist is still probably your best choice to be sure you can get as true of a repair as possible. Then if there is still a descrepency in heights, a self leveler as mentiioned above could be utilized.

this is not from the tub most likely the joists were installed crowning the wrong way and they have always been like that and gotten worse over time. Just put in a drop girder at the worst point to raise up the middle.

How should I support it?