How can I relate $Var(y)$ with $Var(y)_{(i)}$ where $Var(y)_{(i)}$ is de variance of the data with the ith item removed. It is necesary first relate $\bar{y}$ with $\bar{y}_{(i)}$ and it complicates the variance relationship.
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$\begingroup$ Is $y$ a scalar or a vector? And what do you mean with 'item'? $\endgroup$ – Stijn Dec 12 '14 at 19:34
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$\begingroup$ Do you mean sample variance with full dataset versus one observation removed? $\endgroup$ – Adrian Dec 12 '14 at 19:41
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$\begingroup$ Yes, y is the vector $y=(y_1, y_2, ..., y_i,...y_n)$ and $y_{(i)}$ is the vector $y_{(i)}=(y_1, y_2, ..., y_{i-1},y_{i+1}...y_n)$ with the i-th element removed $\endgroup$ – will198 Dec 12 '14 at 19:44
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I think that I have found the solution by myself:
$Var(y)_{(i)}=\frac{n}{n-1}(Var(y)-\frac{(\bar{y}-y_i)^2}{(n-1)})$
Where:
$y_i$ is the i-th observation removed;
n is the number of observations in $y$;
$\bar{y}$ is de mean of $y$
