By John Scaparo, Professional and Certified Canton Michigan Home Inspector.  For more information visit my website.

What appears to be a dark stain on the ceiling is not a water stain or fungal growth. Notice there is no water stain or evidence of moisture on the back side of the drywall in this area above the dark ceiling stain.

The darken area is known as ghosting. Typical ghosting is observed on the ceiling or walls where there is thermal bridging, air leak or defects in the insulation. These conditions create a temperature differential (Cool Spot) that when the moisten-laden air in your home comes in contact with this cooler ceiling or wall surface it condense forming surface dampness. The moist surface then pick up the tiny dust particles in the indoor air and when the area dries out the particles are left over. Over time the dust particles accumulate to form a visible stain like appearance. In the photo notice the darken area on the ceiling. The area is also next to a basement single pane window that was not properly sealed.  There was noticeable air exchange because of the window leak and a considerable temperature drop in this area. On the back side of the drywall there was no water stain that could have caused the ceiling stain. The recommendation to fix this problem is to eliminate or minimize the temperature differential that causes the surface condensation first  by caulking the window air leak or install an insulated type window such as a dual pane or block window.