Estimating standard deviation from wheighted sample

The standard deviation is given by $\sqrt{\frac{\sum ({x}_{i}-x{)}^{2}}{n}}$, however when we estimate the standart deviation from a sample, the best estimation is $\sqrt{\frac{\sum ({x}_{i}-x{)}^{2}}{n-1}}$

How do I have to adjust the standarddeviation if I want to wheight my samples?

I.e. the standard deviation would be $\sqrt{\frac{\sum w({x}_{i})({x}_{i}-x{)}^{2}}{\sum w({x}_{i})}}$, if I had the entire data set. What is the correct estimation of the standard deviation, if I'm only given a subsample of the population?

The standard deviation is given by $\sqrt{\frac{\sum ({x}_{i}-x{)}^{2}}{n}}$, however when we estimate the standart deviation from a sample, the best estimation is $\sqrt{\frac{\sum ({x}_{i}-x{)}^{2}}{n-1}}$

How do I have to adjust the standarddeviation if I want to wheight my samples?

I.e. the standard deviation would be $\sqrt{\frac{\sum w({x}_{i})({x}_{i}-x{)}^{2}}{\sum w({x}_{i})}}$, if I had the entire data set. What is the correct estimation of the standard deviation, if I'm only given a subsample of the population?