How can I compute standard deviation without k observations

Question

For a given variable $X$, we compute the standard deviation. Now I removed $k$ observations from $X_n$ and I would like to compute the new standard deviation $\sigma_{(k)}$ using $\sigma_{n}$.

I found some algorithms to compute the standard deviation on-line for without one observation like this [formula][2]:

$\sigma_n^2=\frac{n-2}{n-1}\sigma^2_{n-1}+\frac{1}{n}(X_n-\bar{X_{n-1}})^2$

But it does not adapt to removing $k$ observations at once.

1- Is there a way to do it for $k$ observations ? (It works for the mean)

2- For which $k$ this is less expensive than computing the standard deviation from scratch?

Answer 1

Comment: Won't fit in comment box.

Use $$S^2 = \frac{1}{n-1}\left[\sum_{i=1}^n X_i^2 - \frac{1}{n}\left(\sum_{i=1}^n X_i\right)^2\right]\\ = \frac{1}{n-1}[Q -T^2/n].$$

Quantities $Q$ and $T$ can be found from $S^2$ and $\bar X.$ Then they can be adjusted to account for removed observations.

Finally, use adjusted $Q^\prime$ and $T^\prime$ to compute the new $S^2.$