# Simplifying the sample variance

Let $Y_1$ and $Y_2$ be i.i.d. R.V.s sampled from a $N(\mu,\sigma^2)$, with $n=2$ it is shown that the sample variance:

$$S^2=\sum_{i=1}^n \frac{(Y_i- \overline Y)^2}{(n-1)}$$ simplifies to

$$S^2=\sum_{i=1}^n (\frac{(Y_1- Y_2)}{\sqrt2})^2$$

I know this must be a simple question, but I do not understand how if $n=2$, the expression get simplified. Substituting $n=2$ for the first expression just gives:

$$S^2=\sum_{i=1}^n {(Y_i- \overline Y)^2}{}$$ so how do we go from here to the second equation?

## EDIT

I expanded the first equation and now I have:

($Y_2^2-2\bar{Y}Y_2+\bar{Y^2})$ + ($Y_1^2-2\bar{Y}Y_1+\bar{Y_2}$) but I am unable to move forward, any hints or tips would be appreciated :)

• What is the definition of $\bar{Y}$? Try plugging that into the first equation. – soakley Jan 27 '14 at 2:28
• Hint: When $n=2$, your expression is $$S^2 = \sum_{i=1}^2 \frac{(Y_i-\bar{Y})^2}{2-1} = (Y_1-\bar{Y})^2 + (Y_2-\bar{Y})^2$$ Now apply soakley's hint of recalling the definition of $\bar{Y}$ and plug this into the two terms on the right of the above equation. – Dilip Sarwate Jan 27 '14 at 4:12