How to determine LVDT sensitivity?
Your math looks right, with the numbers shown.  I suspect that your  measurements, or experimental setup is probably wrong.

Most likely, you are being confused by the AC voltage output measurements.  Try using (-2.48) and (+2.12) in your calculations.  I then get an answer of 4.22 mV/V/mm.

LVDT's tend to have 2 windings, with the signal output (in VAC) going like: 


Distance[mm], Output[Vac]
0 mm,         2.48 Vac
112.5 mm,        0 Vac (right at mid-stroke)
225 mm,       2.12 Vac


So:


dX = 225 mm


and dVac = dVac (1st winding) + dVac (2nd winding), or: 


dVac = 2.48 + 2.12 = 4.6 Vac


So, with Vexcitation = 4.84 Vac,

S = 1000*(dVac / Vexc) / dX:


S = 1000 * (4.6 Vac /4.84 Vac ) / 225 mm = 4.22 [mV/V/mm]