proc=WienerProcess[1,1/2];
data=RandomFunction[proc,{0,5,0.1},25];

{Mean[proc[2.5]],Mean[data[2.5]]}

{Plot[{Mean[proc[t]],Mean[data[t]]},{t,0,5}],Plot[{StandardDeviation[proc[t]],StandardDeviation[data[t]]},{t,0,5}]}

Show[ListLinePlot[data],Plot[{Mean[proc[t]]-3StandardDeviation[proc[t]],Mean[proc[t]]+3StandardDeviation[proc[t]]},{t,0,5},Filling->{1->{2}}]]