Question on weight initialization

In the Weight Initialization section of chapter 3 of the neuralnetworksanddeeplearning book, why does the author say that the the Gaussian distribution is not sharply distributed when half of the input neurons are set to zero?
We start with weight initialization as a standard normal distribution with mean zero and SD = 1. And there are 1000 input neurons, but if half of them are 1s and other half 0s then the SD is sqrt(501) = 22.4 but mean is still 0. Can somebody tell me why is this the case?
If half of random normal numbers are set to zero why should it only impact the SD and not the mean value of the weights?