sig = 5*10^-4;
mn = 162.7;
dat = RandomReal[NormalDistribution[mn, sig], 1000];
Table[
(*function of normal distribution*)
fun = PDF[NormalDistribution[u, s]][x];
mn = Mean[dat];
std = StandardDeviation[dat];
{bin, cnt} = HistogramList[dat, {d std}, "PDF"];
cent = MovingAverage[bin, 2];
cDat = Transpose[{cent, cnt}];
(*find fit using the points*)
sol = fun /. FindFit[cDat, fun, {{u, mn}, {s, std}}, x];
(*use build in function*)
dist = PDF[EstimatedDistribution[dat, NormalDistribution[u, s]]][x];
Show[
h = Histogram[dat, {(d std)}, "PDF"],
ListPlot[cDat],
(*use the range of the histogram for your plot*)
{min, max} = Options[h, PlotRange][[1, 2, 1]];
Plot[{sol, dist}, {x, min, max}]
, ImageSize -> 500]
, {d, .1, .5, .1}]