# Relate $Var(y)$ with $Var(y)_{(i)}$

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.

• Is $y$ a scalar or a vector? And what do you mean with 'item'? – Stijn Dec 12 '14 at 19:34
• Do you mean sample variance with full dataset versus one observation removed? – Adrian Dec 12 '14 at 19:41
• 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 – will198 Dec 12 '14 at 19:44

$Var(y)_{(i)}=\frac{n}{n-1}(Var(y)-\frac{(\bar{y}-y_i)^2}{(n-1)})$
• $y_i$ is the i-th observation removed;
• n is the number of observations in $y$;
• $\bar{y}$ is de mean of $y$