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Let's say a list of numbers $L$ has standard deviation $S$. Is there a formula for finding $S$ if I remove an element $l$ from $L$? Assume we know the mean of both $L$ and $L - l$.

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    $\begingroup$ The question of updating the sample variance as one (or more) new datapoints becomes available is one of online-algorithms. So you can simply take an established algorithm and change it to yield the "old" variance based on the "new" variance, the data point to be removed and supplementary information like the "old" and "new" means. $\endgroup$ Commented Mar 29, 2021 at 9:16
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    $\begingroup$ One (amusing) solution method is to view this as a combination of a dataset of size $n$ and another one of size $-1,$ enabling you to apply (without change) formulas for updating moments when datasets are combined. $\endgroup$
    – whuber
    Commented Mar 29, 2021 at 12:43

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