Maybe I did not explain my issue not well enough.
In the following, I am normalizing the values as its explained in the documentation.
However, I am unable to recreate the actual values obtained when applying bn to beispiel_x
beispiel_x = torch.tensor([[1.,2.,3.,4.,5.],
                           [2.,2.,2.,2.,2.],
                         [3.,3.,3.,3.,3.]])
bn = nn.BatchNorm1d(5)

print((beispiel_x -torch.mean(beispiel_x, axis=0))/torch.sqrt(torch.var(beispiel_x, axis=0)+bn.eps))
print((beispiel_x -bn.running_mean)/torch.sqrt(bn.running_var+bn.eps))
bn(beispiel_x)