$$\sqrt{\frac{\sum_{i = 1}^N{w_i \left( x_i - \overline{x}_w \right)^2}}{\frac{\left( N' - 1 \right) \sum_{i = 1}^N {w_i}}{N'}}}$$
Is it acceptable to remove the $\frac{N' - 1}{N'}$ from the formula of weighted standard deviation?
Since $\frac{N' - 1}{N'}$ is approximately 1. If it is allowed, can you give me a reference. Thank you very much.
The N-1 normalization factor comes from the unbiased estimator of the variance. This makes sure that the expectation of your estimator is the true variance. If you change it by N, the expectation will no longer be exactly the true variance.
However, as N increases, this error becomes of course smaller. But just make sure to take this into account (you can compute the expectation and variance of your estimator) if you want to report the quality or confidence intervals of your estimator.
