I don't know if that fits your theoretical calculation.
I assume you gave standard deviations. Not variances.
I get SD's which are a bit too big but the orders of magnitude seem correct.
Which is not so bad given the simplified model.
I computed how the variance propagates. With the usual
notation: if at a certain ply the acceptance probability
is p then the propagated variance is
sum_i [ (1/p)(x_i^2+s_i^2)-x_i^2 ]
(this assumes the X_i are independent).
Assuming the tree is completely uniform this translates into a linear recursion relation for the variance. I took EBF=perft(10)^(1/10)=30.9
I found the following standard deviations.
0.04% for sampling at ply 6
0.24% for sampling at ply 5
1.48% for sampling at ply 4
The fact that the actual sd's are smaller might indicate that the
tree is not really uniform and that your algorithm adapts to this
non uniformity.