Is Wikipedia wrong about the null distribution of the signed rank statistic? From Wikipedia

Calculate the test statistic $W$ $$
     W = \left|\sum_{i=1}^{N_r} [\operatorname{sgn}(x_{2,i} - x_{1,i}) \cdot R_i]\right|, $$the absolute value of the sum of the signed ranks.
As $N_r$ increases, the sampling distribution of $W$ converges to a
  normal distribution.



*

*Since $W$ is defined to be always  nonnegative, why does the
distribution of $W$ converge to a normal distribution which has
positive probability of negative values?
What should it be then?

*
For $N_r \ge 10$, a z-score can be calculated as $z = \frac{W - 0.5}{\sigma_W}, \sigma_W = \sqrt{\frac{N_r(N_r + 1)(2N_r + 1)}{6}}$.

Does it mean $E(W) = 0.5$ and $\operatorname{Var}(W) = \frac{N_r(N_r + 1)(2N_r +
    1)}{6}$? Why is it true?
Thanks and regards!
 A: While a Normal distribution does always include some positive probability of negative values, that probability can be so small as to be meaningless for all intents and purposes. 
Height of adult humans is usually thought to be normally distributed. Yet the probability of a height less than 0, while positive, is pretty darn small.
W quickly gets to be very large, so it can get very close to a Normal even if it never actually exactly matches one. 
A: Actually, I suspect Wikipedia may be wrong there. That doesn't match the links for starters. That statistic makes me worry for several reasons. The expected value of the sum of the signed ranks of absolute differences will be zero under the null, and a suitably scaled version of it will converge to normality by the CLT. As such, the absolute value of that won't converge to normality, but may, when suitably scaled, converge to a standard half-normal. 
Edit:
Having just simulated the distribution under the null at various sample sizes, it looks like I am right - taking the absolute value of the whole thing makes something that would be asymptotically normal become half-normal. 
I agree - Wikipedia is wrong when it says that statistic converges to normality. It doesn't look like it does, and I don't recall ever having seen the statistic written that way before.
Later edit:
I should show rather than tell: here's an example simulation using the statistic as defined at the Wikipedia page. I did 10000 simulations of signed rank statistics for a continuous distribution symmetric about 0 (representing the differences), at n=100:

