What is the value of a Winchester model 1892 with round barrel made in 1884?

First, your Model 1892 was not made in the year 1884. The Model 1892 was introduced late in the year 1892. The serial number located on the bottom forward edge of the receiver frame will tell us when it was actually made. The "1884" date you found on the gun is the Patent date (John M. Browning invented & patented it). The collector value of your Model 1892 can only be assessed by a visual inspection & evaluation. Old Winchesters are very much like old classic automobiles... the graded condition and factory originality are the key factors in determining what they are truly worth.