= Gaussian absolute-difference expectation
{c}
{title2=$\mathbb E|E_1-E_2|=2\sigma/\sqrt\pi$}
For independent errors $E_1,E_2\sim N(0,\sigma^2)$, their difference is normal with <variance> $2\sigma^2$. Hence $\mathbb E|E_1-E_2|=2\sigma/\sqrt\pi$. To verify the constant, a standard normal variable $Z$ satisfies $\mathbb E|Z|=2\int_0^\infty z e^{-z^2/2}\,dz/\sqrt{2\pi}=\sqrt{2/\pi}$; multiply by $\sqrt2\sigma$. Thus an absolute difference of unbiased noisy ratings has a positive expectation even without a true difference.
Back to article page
