n = 60000; q = 0.495; NSim = 10^5; sigma = Sqrt[4 q (1 - q)];
sims = Table[Total[RandomChoice[{q, 1 - q} -> {1, -1}, n]], {NSim}];
distr = HistogramDistribution[sims];
Show[
 DiscretePlot[PDF[distr, x], {x, -1200, 600}],
 Plot[PDF[NormalDistribution[-600, Sqrt[n]*sigma], x], {x, -1200,
   600}]
 ]